The coefficient-ratio conjecture for eta-derivative polynomials

Let $E_{l,m,N,n}(a)$ denote the coefficient of $y^n$ in the polynomial $E_{l,m,N}(a,y)$ defined by

$$E_{l,m,N}(a,y)=E_{l,m,N,d}(a)y^d+\cdots+E_{l,m,N,0}(a).$$

The coefficient-ratio conjecture. Except for countably many values of $a$, for every positive integer $l$, non-negative integer $m$, and integer $n$ satisfying $0<n\leq m+1$,

$$\frac{n-m-2}{n}\frac{E_{l,m,N,n-1}(a)}{E_{l,m,N,n}(a)}\longrightarrow\eta^{\langle l+m\rangle}(a)$$

as $N\to\infty$.

This is a coefficient-level consequence of the proposed concentration of the roots of $E_{l,m,N}(a,y)$ and is presented with numerical support, but remains unproved.