Relation of gerbes on X×P^N to gerbes on X

Ascertain the correspondence between gerbes with connection on the product orbifold X×P^N and gerbes on X, including how discrete torsion and twisted K-theory data behave under stabilization.

Background

Gerbes classify twists for Chen–Ruan cohomology and twisted K-theory; understanding their behavior under product is key to stable invariants.

The authors ask for an explicit relationship between gerbes on the stabilized orbifold and on the base orbifold.

References

Question To the former classification, can one relate the gerbes over $X\times \bb PN$ with that on $X$?

— A Gromov-Witten approach to $G$-equivariant birational invariants (2405.07322 - Cavenaghi et al., 12 May 2024) in Section 6.4, “Stable G-rationality”