Coefficientwise Hankel-total positivity conjecture for the GKP recurrence

Let $P_n(x;\bm{\mu})$ be the row-generating polynomials of the GKP recurrence, with parameters $x,\alpha,\beta,\gamma,\alpha',\beta',\gamma'$. A sequence of polynomials is coefficientwise Hankel-totally positive when every minor of its Hankel matrix has nonnegative coefficients in the indicated indeterminates.

Coefficientwise Hankel-total positivity conjecture for the GKP recurrence. The sequence

$$\bm{P}=(P_n(x;\bm{\mu}))_{n\geq 0}$$

of row-generating polynomials of the GKP recurrence is coefficientwise Hankel-totally positive, jointly in the seven indeterminates $x,\alpha,\beta,\gamma,\alpha',\beta',\gamma'$.

The conjecture had been confirmed computationally through the $8\times 8$ Hankel matrix $(P_{i+j}(x;\bm{\mu}))_{0\leq i,j\leq 7}$, but no proof or general resolution is given.