Sokal's conjecture on functional-digraph component polynomials

Let $\psi_{n,k}$ be the number of functional digraphs on $[n]$ with $k$ weakly connected components, and define

$$\bm{\psi}_n(y)=\sum_{k=0}^n\psi_{n,k}y^k.$$

A functional digraph is a directed graph in which every vertex has out-degree $1$; a vertex is cyclic if it lies on a directed cycle. Let $[\psi_{n,k}]_{n,k}$ denote the lower-triangular counting matrix.

Sokal's functional-digraph conjecture. The following assertions hold: (i) $[\psi_{n,k}]_{n,k}$ is totally positive; (ii) $(\bm{\psi}_n(y))_{n\geq0}$ is coefficientwise Hankel-totally positive in $y$; (iii) $(\psi_{n+1,1})_{n\geq0}$ is Hankel-totally positive, equivalently, it is a Stieltjes moment sequence.

These assertions concern positivity of functional-digraph enumeration matrices, component polynomials, and the connected case. The source gives no resolution, so the conjecture remains open.