I think you may have misunderstood the abstract of the paper, which says:
Various human activities, including fossil fuel combustion and forest
clearing, emit about eight petagrams (or billion tons) of carbon in
the form of CO2 into the atmosphere annually. The global ocean absorbs
about two petagrams of CO2, and about a half of that amount is
absorbed by the Southern Ocean south of 30°S, thus slowing the rapid
accumulation of CO2 in the atmosphere. Partial pressure of CO2 (pCO2)
is a measure of the chemical driving force for the CO2 exchange
between the ocean and the atmosphere. This paper discusses its space
and time distribution over the Southern Ocean. The major sink zone for
atmospheric CO2 is located in a latitude belt between 30°S and 50°S,
where the biological utilization of CO2 and cooling of warm
subtropical waters flowing southward produce low seawater pCO2.
Note that most of the "half of the global ocean total absorption" occurs in a band between 30°S and 50°S.
winds in this zone also enhance the ocean's uptake. Although the
source-sink conditions vary over a wide range through the seasons in
the areas south of 50°S, this zone is a small sink on an annual
average. Winter observations show that surface water pCO2 values in
the source region for Antarctic Intermediate Water have increased at a
rate faster than the atmospheric increase rate, suggesting that the
ocean CO2 sink intensity has been weakening for several decades and
has changed from a net sink to a net source since 2005. The results of
ocean general circulation-biogeochemistry model studies are found to
be consistent with the observations.*
The "net sink to a net source since 2005" appears to relate to this area south of 50°S, which was only a small sink to begin with. This suggests that the southern ocean as a whole is still a strong carbon sink. As the evidence for an increase in the airborne fraction increasing is currently equivocal, this suggests that the natural carbon sinks (including the southern ocean) is still strengthening in response to increasing atmospheric concentrations.
Note the conclusion of the paper talks of the 30°S and 50°S being a sink in the present tense, which would be inappropriate if it had become a net source since 2005.