The first limiting conjecture for eta-derivative determinants

Let $D_N$ be the determinant polynomial used in the paper, obtained from variables $x_{j,k}$, 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 first limiting conjecture. Except for countably many values of $a$,

$$S_N(a)\longrightarrow 1$$

as $N\to\infty$.

This is the first determinant-ratio specialization of the paper's approximate dependency phenomenon and is supported by numerical data, but no proof or resolution is given.