Minimal-prime classification conjecture for permanental ideals

Let $M_{m,n}=(x_{i,j})$ be a generic $m\times n$ matrix over a field $k$, and let $I_d(m,n)\subset k[x_{i,j}]$ be generated by the permanents of its $d\times d$ submatrices. A minimal prime containing $I_d(m,n)$ is expected, when $\operatorname{char} k>d$ or when $\operatorname{char} k=0$ and $n\geq 2d-1$, to contain either a column of $M_{m,n}$ or all $(d-1)\times(d-1)$ permanents formed from some $m-1$ rows. Minimal-prime classification conjecture. If $\operatorname{char} k>d$ or $\operatorname{char} k=0$ and $n\geq 2d-1$, every minimal prime containing $I_d(m,n)$ must either contain a column of the generic matrix or the $(d-1)\times(d-1)$ permanents of some $m-1$ rows. This conjecture proposes a structural classification of minimal primes over permanental ideals; the supplied text gives no resolution status beyond presenting it as a main conjecture.