  Bilayer graphene (BLG) was recently shown to host a band-inverted phase with
unconventional topology emerging from the Ising-type spin--orbit interaction
(SOI) induced by the proximity of transition metal dichalcogenides with large
intrinsic SOI. Here, we report the stabilization of this band-inverted phase in
BLG symmetrically encapsulated in tungsten-diselenide (WSe$_2$) via hydrostatic
pressure. Our observations from low temperature transport measurements are
consistent with a single particle model with induced Ising SOI of opposite sign
on the two graphene layers. To confirm the strengthening of the inverted phase,
we present thermal activation measurements and show that the SOI-induced band
gap increases by more than 100% due to the applied pressure. Finally, the
investigation of Landau level spectra reveals the dependence of the
level-crossings on the applied magnetic field, which further confirms the
enhancement of SOI with pressure.
