  Strong gate control of proximity-induced spin-orbit coupling was recently
predicted in bilayer graphene/transition metal dichalcogenides (BLG/TMDC)
heterostructures, as charge carriers can easily be shifted between the two
graphene layers, and only one of them is in close contact to the TMDC. The
presence of spin-orbit coupling can be probed by weak antilocalization (WAL) in
low field magnetotransport measurements. When the spin-orbit splitting in such
a heterostructure increases with the out of plane electric displacement field
$\bar D$, one intuitively expects a concomitant increase of WAL visibility. Our
experiments show that this is not the case. Instead, we observe a maximum of
WAL visibility around $\bar D=0$. This counterintuitive behaviour originates in
the intricate dependence of WAL in graphene on symmetric and antisymmetric spin
lifetimes, caused by the valley-Zeeman and Rashba terms, respectively. Our
observations are confirmed by calculating spin precession and spin lifetimes
from an $8\times 8$ model Hamiltonian of BLG/TMDC.
