Canonical-map perfectoidness conjecture for Albanese towers

Let $X/C$ be a smooth proper connected variety of pure dimension $d$ with globally generated $1$-forms, and fix a point $x\in X(C)$. Let

$$H=\prod H_\ell\leq \pi_1(x,X)^{\mathrm{ab}}=T_{\widehat{\mathbb{Z}}}\operatorname{Alb}(X)$$

be a closed subgroup. The canonical map sends a point of $X$ to the $d$-dimensional subspace spanned by the image of the differential of the Albanese morphism. Canonical-map perfectoidness conjecture. The restriction $X_H|_U$ is perfectoid, where $U\subseteq X$ is the preimage under the canonical map

$$X\longrightarrow \operatorname{Gr}_d(\operatorname{Lie}\operatorname{Alb}(X))$$

of the open set of $d$-dimensional subspaces of $\operatorname{Lie}\operatorname{Alb}(X)$ that intersect $H_p\otimes C(-1)$ trivially in $T_p\operatorname{Alb}(X)\otimes C(-1)$. Moreover, no open perfectoid subdiamond of $X_H^\diamond$ has non-empty intersection with the restriction of $X_H^\diamond$ to the complementary closed subvariety of $X$. The conjecture predicts that the canonical map exactly identifies the locus on which the Albanese tower is perfectoid, while ruling out any open perfectoid locus over the complementary closed subvariety. The source gives no evidence of resolution.