Integral existence of the plectic determinant

Prove that the plectic determinant \(D^{\mathrm{plec}}\) exists over the integral universal deformation ring \(R_{\bar D}\), without inverting \(p\), rather than only over \(R_{\bar D}[1/p]\).

Background

The paper constructs a continuous plectic determinant after inverting $p$, obtaining a determinant on $R_{\bar D}[1/p][[G^{\mathrm{plec}}_{F,S}]]$. This rational construction is sufficient for the applications developed in the paper.

The authors explicitly indicate that an integral construction is expected but is not established. The unresolved problem is therefore to remove the inversion of $p$ from the plectic determinant construction.