Total positivity of the Schl"afli--Gessel--Seo polynomial matrix and sequences

For $n,k\geq 0$, let $P_{n,k}(a,b)$ be the Schl"afli--Gessel--Seo forest polynomials, let $P_n(x;a,b)=\sum_{k=0}^n P_{n,k}(a,b)x^k$, and let $P_n(a,b)=P_{n,1}(a,b)$. A polynomial matrix or sequence is coefficientwise totally positive when all its minors have coefficientwise nonnegative polynomial coefficients; a sequence is coefficientwise Hankel-totally positive when all minors of its Hankel matrix have this property. Total positivities for the Schl"afli--Gessel--Seo polynomials. (a) The unit-lower-triangular polynomial matrix $P(a,b)=\bigl(P_{n,k}(a,b)\bigr)_{n,k\geq 0}$ is coefficientwise totally positive jointly in $a,b$. (b) The polynomial sequence ${\bm{P}}=\bigl(P_n(x;a,b)\bigr)_{n\geq 0}$ is coefficientwise Hankel-totally positive jointly in $x,a,b$. (c) The polynomial sequence ${\bm{P}}^\triangle=\bigl(P_{n+1,1}(a,b)\bigr)_{n\geq 0}$ is coefficientwise Hankel-totally positive jointly in $a,b$. These claims extend known total-positivity phenomena for forest triangles and their row-generating polynomials; the source gives no resolution and presents them as conjectures.