Monomial conjecture for maximal permanents of a generic matrix

Let $I_d(d,2d-1)$ be the ideal generated by the maximal permanents of the generic $d\times(2d-1)$ matrix $M_{d,2d-1}$ over a field $k$. Monomial conjecture for maximal permanents. If $\operatorname{char} k>d$ or $\operatorname{char} k=0$, every minimal prime over $I_d(d,2d-1)$ either contains a column of $M_{d,2d-1}$ or contains the $(d-1)\times(d-1)$ permanents of some set of $d-1$ rows. If $\operatorname{char} k\leq d$, a minimal prime may, in addition, contain the $d\times d$ minors of $M_{d,2d-1}$. The text says this conjecture is proved for $d=3$, while the general case is presented as less understood.