The conjectural Fredholm determinant formula for the q-PushASEP q-Laplace transform

Let $q$ satisfy $0<q<1$, let $x_n(t)$ denote the position of particle $n$ at time $t$, and define

$$(a;q)_{\infty}:=\prod_{i=0}^{\infty}(1-aq^i),\qquad (a;q)_k:=\prod_{i=0}^{k-1}(1-aq^i).$$

For all $\zeta\in\mathbb{C}\setminus\mathbb{R}_{>0}$, the conjectural Fredholm determinant formula.

$$\operatorname{\mathbb{E}}\left(\frac{1}{(\zeta q^{x_n(t)+n};q)_{\infty}}\right)=\det(I+K_{\zeta}).$$

Here $\det(I+K_{\zeta})$ is the Fredholm determinant of $K_{\zeta}\colon L^2(C_1)\to L^2(C_1)$, where $C_1$ is a small positively oriented circle containing $1$, and $K_{\zeta}$ has kernel

$$K_{\zeta}(w,w')=\frac{1}{2\pi\mathbf{i}}\int_{-\mathbf{i}\infty+1/2}^{\mathbf{i}\infty+1/2}\frac{\pi}{\sin(-\pi s)}(-\zeta)^s\frac{G(q^s w)}{G(w)}\frac{1}{q^s w-w'}\,ds,$$

with $G(w):=(w;q)_{\infty}^n\Pi_t(w)$. The formula is formally suggested by the moment identities and the rigorously established result when $\mathsf{L}=0$, but is conjectural because the moments grow too rapidly to identify the distribution directly; its validity for the stated q-PushASEP setting remains to be established.