Conjecture on evaluation of prismatic $G$-$F$-gauges

Let $R$ be a complete regular local ring over $W(k)$ with residue field $k$, let $(\mathfrak{S},(\mathcal{E}))=(W(k)[[t_1,\dotsc,t_n]],(\mathcal{E}))$ be a prism of Breuil--Kisin type with $R\simeq\mathfrak{S}/\mathcal{E}$ over $W(k)$, and let $\mu$ be 1-bounded. For a perfectoid pair $(S,a^\flat)$ over $W(k)$, use the associated prismatic $G$-$F$-gauges and displays. Evaluation conjecture for prismatic $G$-$F$-gauges. If $\dim R=1$, then for every integer $m\geq1$ the natural functor

$$G\mathchar`-F\mathchar`-\mathrm{Gauge}_\mu(R/\mathfrak{m}_R^m)\to G\mathchar`-\mathrm{Disp}_\mu(\mathfrak{S}_m,(\mathcal{E}))$$

is an equivalence; and for every perfectoid pair $(S,a^\flat)$ and every integer $m\geq1$, the natural functor

$$G\mathchar`-F\mathchar`-\mathrm{Gauge}_\mu(S/a^m)\to G\mathchar`-\mathrm{Disp}_\mu(W(S^\flat)/[a^\flat]^m,I_S)$$

is an equivalence. These assertions are conjectural in general; the paper notes that related statements should follow from work of Gardner--Madapusi, and records cases known for $G=\operatorname{GL}_N$.