The Taylor-coefficient limiting ratio conjecture for eta derivatives

Let $D_{!,N}$ be the determinant polynomial obtained by replacing each variable $x_{j,k}$ in $D_N$ by $x_{j,k}/k!$, and define

$$S_{!,N}(a)=\frac{D_{!,N}\big(x_{j,k}\mapsto\eta^{\langle j+k\rangle}(a)\big)}{D_{!,N-1}\big(x_{j,k}\mapsto\eta^{\langle j+k+2\rangle}(a)\big)}.$$

The Taylor-coefficient limiting ratio conjecture. Except for countably many values of $a$,

$$S_{!,N}(a)\longrightarrow 1$$

as $N\to\infty$.

This modification uses factorially normalized, Taylor-like coefficients and is reported to converge more slowly than the first limiting ratio; it remains conjectural.