The toric Fano $\mathrm{ch}_2$-positivity conjecture

Let $X$ be a smooth toric Fano $d$-fold. A smooth toric Fano variety is $\mathrm{ch}_2$-positive when its second Chern character has positive intersection with every relevant surface, as in the paper. The toric Fano $\mathrm{ch}_2$-positivity conjecture. If $X$ is $\mathrm{ch}_2$-positive, then $X$ is isomorphic to the $d$-dimensional projective space $\mathbb{P}^d$. The preceding results establish this in the dimensions and Picard-number cases covered by the paper, while the general assertion is proposed as a conjecture.