The minor-ratio conjecture for eta-derivative determinants

Let $\mathfrak D=d_0,d_1,\dots$ and $\mathfrak F=f_0,f_1,\dots$ be strictly increasing sequences of non-negative integers, and let

$$D_{N,\mathfrak D,\mathfrak F}=\det\big(x_{f_i,d_j}\big)_{0\leq i,j<N}.$$

Let $\mathfrak D_1=d_1,d_2,\dots$ and $\mathfrak F_1=f_1,f_2,\dots$ be their tails, and let $\mathfrak M_{\mathfrak D}$ and $\mathfrak M_{\mathfrak F}$ be the sets of non-negative integers omitted by the respective sequences.

The minor-ratio conjecture. If $d_0=f_0=0$ and both omitted sets are finite, then, except for countably many values of $a$,

$$S_{N,\mathfrak D,\mathfrak F}(a)=\frac{D_{N,\mathfrak D,\mathfrak F}\big(x_{j,k}\mapsto\eta^{\langle j+k\rangle}(a)\big)}{D_{N-1,\mathfrak D_1,\mathfrak F_1}\big(x_{j,k}\mapsto\eta^{\langle j+k+2\rangle}(a)\big)}\longrightarrow 1$$

as $N\to\infty$.

This extends the basic determinant-ratio conjecture to minors determined by sequences with finitely many missing indices; the supplied text gives no resolution beyond the stated conjectural extension.