The uniform non-modularity conjecture for the Hurwitz-space construction

Let $p$ be a prime and let $\mathcal H(\widetilde F_{2,2}/\Phi^1\times^s J_3,\mathbf C_{\pm3^2})^{\mathrm{abs},\mathrm{rd}}$ be the absolute reduced Hurwitz space in the indicated level-zero construction. The conclusion of the cited proposition is that this space embeds in $\mathbb P^1_j\times\mathbb P^1_j$ but is not a modular curve.

Uniform non-modularity conjecture. The same conclusion holds for every $p\neq 3$.

The assertion extends the level-zero non-modularity result from $p=3$ to all other primes in the construction. The source gives no evidence of resolution in the excerpt.