Is it possible to measure new general relativistic third-body effects on the orbit of Mercury with BepiColombo?

Recently, Will calculated an additional contribution to the Mercury’s precession of the longitude of perihelion $$\varpi $$ of the order of $${\dot{\varpi }}_\text {W}\simeq 0.22~\text {milliarcseconds per century}$$ ($$\text {mas cty}^{-1}$$). It is partly a direct consequence of certain 1pN third-body accelerations entering the planetary equations of motion, and partly an indirect, mixed effect due to the simultaneous interplay of the standard 1pN pointlike acceleration of the primary with the Newtonian N-body acceleration, to the quadrupole order, in the analytical calculation of the secular perihelion precession with the Gauss equations. We critically discuss the actual measurability of the mixed effects with respect to direct ones. The current uncertainties in either the magnitude of the Sun’s angular momentum $$S_\odot $$ and the orientation of its spin axis $${\varvec{\hat{S}}}_\odot $$ impact the precessions $${\dot{\varpi }}_{J_2^\odot },~{\dot{\varpi }}_\text {LT}$$ induced by the Sun’s quadrupole mass moment and angular momentum via the Lense–Thirring effect to a level which makes almost impossible to measure $${\dot{\varpi }}_\text {W}$$, even in the hypothesis that it comes entirely from the aforementioned 1pN third-body accelerations. On the other hand, from the point of view of the Lense–Thirring effect itself, the mismodeled quadrupolar precession $$\delta {\dot{\varpi }}_{J_2^\odot }$$ due to the uncertainties in $${\varvec{\hat{S}}}_\odot $$ corresponds to a bias of $$\simeq 9\%$$ of the relativistic one. The resulting simulated mismodeled range and range-rate times series of BepiColombo are at about the per cent level of the nominal gravitomagnetic ones