Primality conjecture for permanental ideals

Let $X$ be an $m\times n$ generic, symmetric, or Hankel matrix, and let $\Bbbk$ be a field of characteristic $p\neq 2$. Write $P_t(X)$ for the ideal generated by the $t\times t$ permanents of $X$. Primality conjecture for permanental ideals. The ideal $P_t(X)$ of $\Bbbk[X]$ is prime if and only if $t=m=n$. The conjecture is supported by the known cases for $t=2$ and $t=3$, but remains open in general.