Stable G-birational invariance of H^1(G;Pic_torsion(𝒢1))

Determine whether the group H^1(G; Pic_{torsion}(𝒢1))—the first cohomology of G with coefficients in the Picard subgroup of gerbe-induced line bundles on the arrow space 𝒢1—is a stable G-birational invariant.

Background

Earlier, the authors proved a birational invariance for the Picard subgroup arising from gerbes over Deligne–Mumford stacks.

Here they ask whether the corresponding torsion Picard cohomology is invariant under stabilization by projective space.