Weak Hom-version conjecture for ordinary moduli spaces of curves

Let $M_{g,n}^{\rm ord}$ be the ordinary moduli space of smooth pointed stable curves, and let $M_{g,n,\mathbb{F}_p}$ be its characteristic-$p$ counterpart. 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 the closure of $q_i$ and let $\pi_1^{\rm t}(q_i)$ denote the corresponding tame fundamental group. Weak Hom-version conjecture.

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

is non-empty if and only if $V_{q_{1}}\supseteq_{fe}V_{q_{2}}$ in the ordinary case, respectively $V_{q_{1}}\supseteq V_{q_{2}}$ in characteristic $p$. The conjecture is intended to recover the topological structures of the moduli spaces from tame fundamental groups; the source reports no published results for non-closed points.