Hankel total positivity conjecture for vincular pattern Classes 1 and 2

For each $n\geq 0$, let $P_n^{(1)}(x)$ and $P_n^{(2)}(x)$ be the generating polynomials for the number of occurrences of vincular patterns in Class $1$ and Class $2$, respectively. A polynomial sequence is coefficientwise Hankel totally positive in $x$ when every minor of its Hankel matrix has coefficients that are nonnegative polynomials in $x$.

Hankel total positivity conjecture. The sequences of polynomials

$$\left(P_n^{(1)}(x+1)\right)_{n\geq 0},\qquad \left(P_n^{(2)}(x+1)\right)_{n\geq 0}$$

are both coefficientwise Hankel totally positive with respect to the variable $x$.

The unshifted sequences are not coefficientwise Hankel totally positive, while computations suggest that the shift $x\mapsto x+1$ restores this stronger positivity property. The claim is supported by finite tests but remains open.