Tamagawa's weak Isom-version conjecture for ordinary moduli spaces

Let $M_{g,n}$ be the ordinary moduli space of smooth pointed stable curves and let $M_{g,n,\mathbb{F}_p}$ be the corresponding characteristic-$p$ moduli space. For arbitrary points $q_i\in M_{g,n}$, respectively $q_i\in M_{g,n,\mathbb{F}_p}$, $i\in\{1,2\}$, let $V_{q_i}$ denote their closures and let $\pi_1^{\rm t}(q_i)$ denote the tame fundamental groups of the associated punctured curves. Tamagawa's weak Isom-version conjecture.

$${\rm Isom}_{\rm pg}(\pi_{1}^{\rm t}(q_{1}),\pi_{1}^{\rm t}(q_{2}))$$

is non-empty if and only if $V_{q_{1}}=_{fe}V_{q_{2}}$ in the ordinary case, respectively $V_{q_{1}}=V_{q_{2}}$ in characteristic $p$. The conjecture is presented as a consequence of the weak Hom-version conjecture, while the source says that both conjectures remain unresolved for non-closed points.