Conjecture on the canonical equivalence for coherent overconvergent F-isocrystals

Let $X$ be a $k$-variety, let $Y$ be a smooth open of $X$, and suppose that $(Y,X)$ is $d$-embeddable. Denote by $(F\text{-})D^\mathrm{b}(\operatorname{Coh}(X,\mathcal{P},T))$ the bounded derived category of coherent objects and by $(F\text{-})D^\mathrm{b}_\mathrm{coh}(\mathcal{P},T,X/K)$ the corresponding category of coherent arithmetic $F$-objects.

Canonical equivalence conjecture. The canonical functor

$$(F\text{-})D^\mathrm{b}(\operatorname{Coh}(X,\mathcal{P},T))\to (F\text{-})D^\mathrm{b}_\mathrm{coh}(\mathcal{P},T,X/K)$$

should be an equivalence of categories.

If true, the properness or open-immersion hypothesis on the morphism $f$ would be unnecessary in the preceding direct-image result, and the corresponding categories could be defined by replacing “properly $d$-embeddable” with “$d$-embeddable.” The source gives no resolution of this conjecture.