The probability of Simpson conversion in four-dimensional binary tables

Let $p_{ijk\ell}$ be entries of a $2\times2\times2\times2$ table sampled uniformly from the probability simplex, with $i,j,k,\ell\in\{0,1\}$ and $p_{ijk\ell}\geq0$. Consider the two $2\times2\times2$ subtables obtained by fixing $\ell=0$ and $\ell=1$, and their sum over $\ell$. A Simpson conversion occurs when the two subtables induce the same triangulation of the cube, while their sum induces a different triangulation. The probability conjecture. The probability that a Simpson conversion occurs in this context is $1/450$. This is based on computational sampling and is presented as a conjecture about the frequency of Simpson conversions in $2\times2\times2\times2$ contingency tables; the corresponding two-dimensional probability is known to be $1/60$.