An article on this topic has recently been published: Elastic interaction between Mauna Loa and Kīlauea evidenced by independent component analysis (Przeor et al., 2022). It begins:
The interaction processes between the two most active Hawaiian
volcanoes are still controversial, and despite multiple studies
carried out over more than a century, an unambiguous model has yet to
be identified.
Later, the two following paragraphs review some of those studies:
The interaction between the two most active volcanoes of Hawai‘i
Island has been discussed for over 100 years11,12,26. Rhodes and
Hart27 confirm that the chemical composition of lavas at Kīlauea and
Mauna Loa are different, indicating the magmatic feeding systems are
independent (at least at the crustal level). However, geophysical
studies seem to indicate the opposite: Klein11 first noticed the
anticorrelation between the two volcanoes, emphasising that an
increase in activity at Kīlauea often corresponds to a decrease in
Mauna Loa dynamism. Miklius and Cervelli12 captured the opposite
behaviour in the ground deformation patterns of the volcanoes: at the
beginning of the high-volume effusive episode in Kīlauea, inflation of
Mauna Loa was observed (May 2002). Shirzaei et al.28 studied the
coupling behaviour of both volcanoes between 2003 and 2008. The
authors postulate that the causative source of the interaction between
the two volcanoes is related to deep-seated mantle surges. Despite
being the subject of many studies, the nature and the mechanism of the
interaction between the feeding systems of the two volcanoes and their
level of interconnection are still contentious and remain unclear.
Many authors have already suggested possible connections between Mauna
Loa and Kīlauea. Miklius and Cervelli12 proposed a crustal-level
interaction between them: pulses of magma in the plumbing system of
Mauna Loa may cause pressure variations in the Kīlauea shallow magma
system. Gonnermann et al.13 explained observed related ground
deformation patterns by considering a pore-pressure diffusion within a
thin accumulation layer in the asthenosphere. Since the magma
composition at Kīlauea is isotopically distinct from Mauna Loa27,
Gonnermann et al.13 suggested the interconnection between Mauna Loa
and Kīlauea must be explained by the transfer of stress by
pore-pressure variations. Shirzaei et al.28 explain the
interconnection between the Mauna Loa magma chamber and the Kīlauea
rift zone through pore pressure diffusion in an asthenospheric magma
supply system.
This new study, based on deformation patterns of both Mauna Loa and Kīlauea, concludes:
We highlight that the connection between Mauna Loa and Kīlauea occurs at shallow depths in the first few kilometers of the crust, through a stress transfer mechanism. [...] This interconnection is created by the Mauna Loa reservoir perturbing the Kīlauea shallowest source. Conversely, the sources below the Kīlauea do not effectively influence the Mauna Loa reservoir. In practice, the inflation of Mauna Loa makes the stress field in the surroundings of Kīlauea less favourable for the ascent of magma into its shallow reservoir.
In summary, according to this new study:
- There is a "connection" between the two volcanoes (or rather between their plumbing systems), but this connection works in one way only$-$Mauna Loa has an influence on Kīlauea, but the opposite is not true.
- This is not a physical connection of the plumbing systems (as in shared pipes), but rather a transfer of stress through the crust.
- This connection is actually an anticorrelation: when there is an overpressure in Mauna Loa magma reservoir (hence inflation of the volcano), there is a decrease in stress in Kīlauea's shallower reservoir.
- But! Kīlauea has another, deeper reservoir which seems to be unaffected by this connection. Note that this reservoir is also connected to Kīlauea rift zones, so its magma can erupt without going through the upper reservoir.
All in all, I'd say that each volcano works pretty much on its own, independently, or, to answer your question directly, that there is no causal relationship between the simultaneous eruptions of Mauna Loa and Kīlauea. But we still don't know for sure. Despite being very well studied for more than a century, Kīlauea still holds some mysteries! The geometry of its plumbing system is not so well defined yet. The USGS carried out an Airborne Electromagnetic and Magnetic (AEM) survey this summer precisely to better constrain it, so we might have some answers soon!
Telephoto image of the Airborne Electromagnetic and Magnetic (AEM) survey being flown over Kīlauea caldera on July 28, 2022. The South Sulphur Bank area is visible in the background of the image as a light-colored deposits on the left side. NPS Photo/J.Wei, public domain.
Update. Both eruptions have now ended, at about the same time. Here is a USGS statement about this situation:
Mauna Loa and Kīlauea have separate magmatic systems and are not connected at a shallow level. But it remains possible that the volcanoes can influence one another through stresses associated with their eruptions—essentially, when one volcano expands or contracts, it can put pressure on (or take pressure off!) the other volcano. [...] Removal of magma from Mauna Loa’s reservoir by the recent eruption may have allowed Kīlauea to relax. Because Kīlauea’s eruption was occurring at such a low rate, it could have been more susceptible to the small changes in stress that were caused by Mauna Loa’s eruption. However, it is also possible that the change in Kīlauea’s eruptive behavior is a coincidence—with such low levels of activity at Kīlauea prior to Mauna Loa’s eruption, it is difficult to be certain. And of course this may only represent a temporary pause in Kīlauea’s activity.
Update #2. Two new studies have been published on the subject of "connected" volcanoes (although not in Hawaii). Kiryukhin et al. (2023) describes how Bezymianny eruptions are actually fed by nearby Klyuchevskoy magma chamber. Reddin et al. (2023) shows that six (!) volcanoes from the Galápagos Islands are connected through interconnected magmatic systems.