The delta-process tree probability conjecture

Let $k$ and $r$ be positive integers with $r<k$, let $\textbf{p}=(p_1,\ldots,p_k)$ be a tuple of non-negative integers, and let $\mathcal{R}_{\textbf{p},r}$ be the specified set of $r$-tuples of subsets. Suppose this set is non-empty. Choose $\textbf{S}=(S_1,\ldots,S_r)$ uniformly from $\mathcal{R}_{\textbf{p},r}$ and choose a surjection $f:[k-1]\to[r]$ uniformly and independently of $\textbf{S}$. Let $G_\delta(\textbf{S},f)$ be the random digraph defined by the delta rule, and write $P_\delta(\textbf{p},r)$ for the probability that it is a tree.

The delta-process tree probability conjecture. One has

$$P_\delta(\textbf{p},r)=\frac{|\mathcal{R}_{\textbf{p},r-1}|}{|\mathcal{R}_{\textbf{p},r}|}.$$

This is equal to the probability that $S_1=\emptyset$.

The paper states that this conjecture suggests a simple expression for the tree probability and reports that it could not be proved. It is equivalent there to an identity involving a generalized probabilistic determinant.