Chan–Leung's facet-point conjecture for reflexive polytopes

Let $P$ be a reflexive polytope with barycenter $b_P=0$. For a facet $F$ of $P$, let a facet $G$ be adjacent to $F$ when there is a vector $u_G$ such that

$$\langle u_G,G\rangle=-1,\qquad \langle u_G,P\rangle\geq -1.$$

Chan–Leung's facet-point conjecture. If $b_P=0$, then for every facet $F$ there exists a point $x_F\in\operatorname{aff}(F)$ such that

$$\langle u_G,x_F\rangle\leq\frac12$$

for every facet $G$ of $P$ adjacent to $F$.

The conjecture was proposed as a combinatorial ingredient for proving the Chan–Leung Chern-number inequality without additional assumptions. The source gives a five-dimensional counterexample, so the assertion is refuted.