Sokal's Hankel-total positivity conjecture for GKP row-generating polynomials

Let $P_n(x;\bm{\mu})$ be the row-generating polynomials of the Graham–Knuth–Patashnik recurrence, with parameters $\bm{\mu}=(\alpha,\beta,\gamma,\alpha',\beta',\gamma')$ treated as indeterminates. Thus the coefficientwise partial order is taken in the seven variables $x,\alpha,\beta,\gamma,\alpha',\beta',\gamma'$ on ${\mathbb Z}[x,\bm{\mu}]$. A sequence $(a_n)_{n\geq 0}$ is Hankel-totally positive when every minor of its Hankel matrix $(a_{i+j})_{i,j\geq 0}$ is nonnegative in this order. Sokal's conjecture. The sequence $(P_n(x;\bm{\mu}))_{n\geq 0}$ of row-generating polynomials of the GKP recurrence is coefficientwise Hankel-totally positive, jointly in all seven indeterminates $x$ and $\alpha,\beta,\gamma,\alpha',\beta',\gamma'$. This conjecture concerns total positivity of the Hankel matrix formed from the GKP row-generating polynomials and remains open.