Chan–Leung's Miyaoka–Yau inequality for Kähler–Einstein toric Fano manifolds

Let $X$ be a Kähler–Einstein toric Fano $n$-fold, and let $H$ be a nef class on $X$. Chan–Leung's conjecture.

$$c_1^2(X)H^{n-2} \leq 3c_2(X)H^{n-2}.$$

Chan and Leung proved this in dimensions $n=2,3,4$ and when every facet of the associated reflexive polytope contains an interior lattice point; the inequality was checked computationally for $n\leq 7$ in the anticanonical case. The general assertion remains open in the source.