The probabilistic determinant formula for the delta process

Let $D\subseteq[k-1]$, and let $M^D$ be the matrix defined in the paper from the delta-process indicator variables. Let $\mathbb{P}\!\det(M^D)$ denote its probabilistic determinant, let $\mathcal{R}_{\textbf{p},r}$ and $\mathcal{S}_{\textbf{p},r}$ be the corresponding sets of tuples, and choose $\textbf{S}$ uniformly from $\mathcal{R}_{\textbf{p},r}$.

The probabilistic determinant formula. For every $D\subseteq[k-1]$,

$$\mathbb{P}\!\det(M^D)=\frac{|\mathcal{R}_{\textbf{p},r}|}{|\mathcal{S}_{\textbf{p},r}|}\times\mathbb{P}\bigl(k\notin S_1,\ D\subseteq S_1\mid \textbf{S}\in\mathcal{R}_{\textbf{p},r}\bigr).$$

The paper presents this as a formula suggested by calculations and as a possible route toward proving the preceding delta-process tree probability conjecture; no proof is given.