Total positivity of bivariate forest polynomials

Let $f^\sharp_{n,k}(w)$ be the entries of the exponential Riordan array with generating function $G(t)=(e^{wT(t)}-1)/w$, where $T(t)$) is the tree function, and define $F_n^\sharp(x,w)=\sum_{k=0}^n f^\sharp_{n,k}(w)x^k$. A polynomial matrix is coefficientwise totally positive when all minors have coefficientwise nonnegative coefficients; a polynomial sequence is coefficientwise Hankel-totally positive when all Hankel minors have coefficientwise nonnegative coefficients. Total positivities for the bivariate forest polynomials. (a) $F^\sharp=\bigl(f^\sharp_{n,k}(w)\bigr)_{n,k\geq 0}$ is coefficientwise totally positive in $w$. (b) ${\bm{F}}^\sharp=\bigl(F_n^\sharp(x,w)\bigr)_{n\geq 0}$ is coefficientwise Hankel-totally positive in $x,w$. (c) ${\bm{F}}^{\sharp\triangle}=\bigl(f^\sharp_{n+1,1}(w)\bigr)_{n\geq 0}$ is coefficientwise Hankel-totally positive in $w$. The source verifies parts (a) and (b) only to finite orders and notes that (c) follows from (b); no general proof is given.