The shifted generalized forest submatrix Hankel-total positivity conjecture

Let $G'(p,q,\gamma,z)=\mathsf{F}(p,q,0,0,\gamma,0,0,z)\operatorname{diag}(((pq\gamma)^{n(n-1)/2})_{n\geq0})$, and let $G'_n(x;p,q,\gamma,z)=\sum_{k=0}^n g'_{n,k}(p,q,\gamma,z)x^k$. The shifted generalized forest submatrix conjecture. The polynomial sequence $(G'_n(x;p+1,q+1,\gamma+1,z))_{n\geq0}$ is coefficientwise Hankel-totally positive jointly in $p,q,\gamma,z,x$. The unmodified two-parameter specialization is not even $\mathrm{TP}_2$, while the diagonal modification and unit shifts are supported by computation; the conjecture remains open.