Universality conjecture for deformations of prismatic $G$-$F$-gauges

Let $k$ be the residue field, let $\mu$ be 1-bounded, and let $\mathscr{Q}\in G\mathchar`-F\mathchar`-\mathrm{Gauge}_\mu(k)$. Let $R_{G,\mu}$ represent the deformation functor on complete regular local rings over $W(k)$, and let $\mathscr{Q}^{\mathrm{univ}}$ be the corresponding universal deformation. For a complete noetherian local quasisyntomic ring $R$ over $W(k)$ with residue field $k$, let $\operatorname{Hom}(R_{G,\mu},R)_e$ denote the set of local homomorphisms over $W(k)$. Universality conjecture for prismatic $G$-$F$-gauge deformations. The map

$$\operatorname{Hom}(R_{G,\mu},R)_e\to\operatorname{Def}(\mathscr{Q})_R$$

induced by $\mathscr{Q}^{\mathrm{univ}}$ is bijective. Moreover, for every perfectoid pair $(S,a^\flat)$ over $W(k)$, after viewing $\mathscr{Q}$ over $S/a$ by base change, the map

$$\operatorname{Hom}(R_{G,\mu},S/a^m)_e\to\operatorname{Def}(\mathscr{Q})_{S/a^m}$$

is bijective for every $m\geq1$, where the target is the set of isomorphism classes of deformations in $G\mathchar`-F\mathchar`-\mathrm{Gauge}_\mu(S/a^m)$. This proposes extension of the universal deformation beyond complete regular local rings; the general assertion remains open, with known special cases for $G=\operatorname{GL}_N$.