F-purity conjecture for the $3\times4$ third permanental ideal

Let $X$ be a $3\times 4$ generic matrix of indeterminates, and let $\Bbbk$ be a field of characteristic $p>2$. F-purity conjecture for the $3\times4$ third permanental ideal. The quotient ring $\Bbbk[X]/P_3(X)$ is $F$-pure if and only if

$$p\equiv 1 \pmod{6}.$$

The authors verified this computationally for characteristics at most $31$. The conjecture concerns the first unresolved case beyond the established $t=2$ result; the analogous $P_4(X_{4\times5})$ case is left speculative rather than asserted as a conjecture.