Determinantal presentation conjecture for the transfer elimination ideal

Let $n=qp$ and let $I=\langle f_0,f_1,\ldots,f_{p-1}\rangle\cap R^{S_n}$ be the elimination ideal, where the $f_i$ are the polynomials obtained by collecting the coefficients of the characteristic polynomial by powers of $t$ modulo $p$. Let $A=[A_0\ A_1\ \cdots\ A_{p-1}]$ be the $(2q-1)\times((q-1)+(p-1)q)$ block matrix formed from the Sylvester blocks $A_i$, and let $J=\mathcal{I}_{2q-1}(A)$ be its ideal of maximal minors. Determinantal presentation conjecture. The elimination ideal $I$ is equal to the determinantal ideal $J$. The supplied context says that the paper proves only a radical containment in general and proves the equality in the case $q=2$, so the conjecture remains open in the full stated generality.