Subdivision conjecture for the MTP2 log-concave maximum likelihood estimator

Let $X\subseteq\mathbb{R}^d$ be the sample configuration, let $X'\supseteq X$ and let $\Delta'$ be the finite set and bimonotone subdivision produced by the paper's algorithm. Let $h_{X',y}$ denote the tent function on $X'$ with heights $y$, and let $\mathcal S$ be the feasible height set. Subdivision conjecture. The MTP2 log-concave MLE is a piecewise-linear function whose subdivision is $\Delta'$ or any subdivision refining $\Delta'$. Equivalently, if $\hat y$ solves the stated optimization problem with the original sample weights on $X$ and weight zero on $X'\setminus X$, then $h_{X',\hat y}$ is supermodular. This is the proposed algorithmic description of the MTP2 MLE for non-tidy configurations and remains open.