Total positivity of bivariate functional-digraph polynomials

Let $\Psi_n(x,y)$ be the generating polynomial for functional digraphs on $[n]$, with weight $x$ for each cyclic vertex and weight $y$ for each component. Define $\psi^{\rm X}_{n,k}(y)=[x^k]\Psi_n(x,y)$ and $\psi^{\rm Y}_{n,k}(x)=[y^k]\Psi_n(x,y)$. 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 functional-digraph polynomials. (a) The unit-lower-triangular matrix $\Psi^{\rm Y}=\bigl(\psi^{\rm Y}_{n,k}(x)\bigr)_{n,k\geq 0}$ is coefficientwise totally positive in $x$. (b) The sequence ${\bm{\Psi}}=\bigl(\Psi_n(x,y)\bigr)_{n\geq 0}$ is coefficientwise Hankel-totally positive in $x,y$. (c) ${\bm{\Psi}}^{{\rm Y}\triangle}=\bigl(\psi^{\rm Y}_{n+1,1}\bigr)_{n\geq 0}$ is coefficientwise Hankel-totally positive in $x$. The claims extend the known positivity of the other coefficient matrix and its one-component column; the source gives no resolution.