Strong coefficientwise Hankel-total positivity conjecture for the GKP recurrence

Let $T(n,k)$ be defined by the shifted GKP recurrence

$$T(n,k)=[\widetilde{\alpha}(n-1)+\widetilde{\beta}k+\widetilde{\gamma}]T(n-1,k)+[\widetilde{\alpha}'(n-1)+\widetilde{\beta}'(k-1)+\widetilde{\gamma}']T(n-1,k-1),$$

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

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

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

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

This is presented as a stronger version of the unshifted GKP positivity conjecture, obtained by an affine reparametrization, and remains unproved in the supplied text.