Conjecture on the equivalence induced by the functor $D_X$

Let $R$ be a $p$-adically complete small $\mathcal O_K$-algebra, and let $X=\operatorname{Spf}(R)$. Consider the functors

$$D_X: \mathrm{Vect}(X_{\mathbbl{\Delta}}^\circ,\mathcal O_{\mathcal E,\mathbbl{\Delta}}^{\dagger})^{\varphi=1}\to \mathrm{Vect}(X_{\mathbbl{\Delta}},\mathcal O_{\mathcal E,\mathbbl{\Delta}})^{\varphi=1}$$

and

$$D_X^{\operatorname{perf}}: \mathrm{Vect}(X_{\mathbbl{\Delta}}^{\circ,\operatorname{perf}},\mathcal O_{\mathcal E,\mathbbl{\Delta}}^{\dagger})^{\varphi=1}\to \mathrm{Vect}(X_{\mathbbl{\Delta}}^{\operatorname{perf}},\mathcal O_{\mathcal E,\mathbbl{\Delta}})^{\varphi=1}.$$

Equivalence conjecture. The functors $D_X$ and $D_X^{\operatorname{perf}}$ each define an equivalence of categories. This conjecture predicts that the comparison between the overconvergent and ordinary prismatic categories is categorical, both on the ordinary and perfect transversal prismatic sites. The supplied text does not indicate whether the conjecture has been proved or disproved.