Eigenvalue conjecture for the degenerate elliptic operator

Let $D_{n,y}^1(p)$ be the Ruijsenaars difference operator acting on formal Laurent power series in $y$, and let $\big[\eta(z;p q^{-1}t)\big]_1$ act on the Fock space $\mathcal{F}$. For a partition $\lambda$, write $\varepsilon_\lambda(p,n)$ for the eigenvalue determined by

$$D_{n,y}^1(p)f_\lambda(y,p)=\varepsilon_\lambda(p,n)f_\lambda(y,p).$$

Eigenvalue conjecture. The eigenvalues of $\big[\eta(z;p q^{-1}t)\big]_1$ on $\mathcal{F}$ are given by

$$\lim_{n\rightarrow\infty}(1-t^{-1})\frac{(p/t;p)_\infty(p t/q;p)_\infty}{(p;p)_\infty(p/q;p)_\infty}t^{-n+1}\varepsilon_\lambda(p,n),$$

where $\lambda$ ranges over partitions. Equivalently, the unwanted operator satisfies

$$\lim_{n\rightarrow\infty}t^{-n}\left[\prod_{i=1}^n\frac{\Theta_p(q t^{-1} z/y_i)}{\Theta_p(q z/y_i)}\frac{\Theta_p(t z/y_i)}{\Theta_p(z/y_i)}\eta(z;p q^{-1}t)\right]_1\cdot 1=0.$$

This conjecture identifies the limiting eigenvalues of the degenerate elliptic operator with those obtained from the finite-variable Ruijsenaars operators. The source presents it as suggested by brute-force calculations and does not provide a resolution.