The root-concentration conjecture for eta-derivative polynomials

Let $E_{l,m,N}(a,y)$ be the polynomial defined by specializing the determinant $D_N$ so that $x_{j,k}=y$ when $j+k=m$, and $x_{j,k}=\eta^{\langle l+j+k\rangle}(a)$ otherwise. Let its roots be counted with multiplicity.

The root-concentration conjecture. Except for countably many values of $a$, for every positive integer $l$ and non-negative integer $m$, the maximal distance between the roots of $E_{l,m,N}(a,y)$ and $\eta^{\langle l+m\rangle}(a)$ tends to zero as $N\to\infty$.

The conjecture concerns the stronger root-level form of the limiting phenomenon; the paper reports numerical evidence for related cases and derives corollaries, but leaves this assertion unresolved.