The shifted generalized forest-matrix Hankel-total positivity conjecture

Let $\tilde{\mathsf{F}}(q,y,z)$ be the generalized forest matrix, and define $\bar{\mathsf{F}}(q,y,z)=\tilde{\mathsf{F}}(q,y,z)\operatorname{diag}((q^{k(k-1)/2})_{k\geq0})$. Let $\bar{\mathsf{F}}_n(x;q,y,z)=\sum_{k=0}^n\bar{f}_{n,k}(q,y,z)x^k$, with $\bar{\mathsf{F}}_0(x;q,y,z)=1$. The shifted generalized forest-matrix conjecture. The Hankel matrix

$$H_{\infty}(\bar{\mathsf{F}}_{n+n'}(x;q,y,z))_{n,n'\geq0}$$

is coefficientwise totally positive jointly in $q,x,y,z$. The unshifted generalized sequence fails coefficientwise total positivity, whereas computations up to $5\times5$ suggest the diagonal modification restores it; the conjecture remains open.