The toric Fano surface-intersection conjecture

Let $X$ be a smooth toric Fano $d$-fold, and let $V^d$ denote the exceptional toric Fano variety used in the paper. A torus invariant subsurface is a torus-invariant surface $S\subset X$ of Picard number two. The toric Fano surface-intersection conjecture. If $X$ is isomorphic to neither $\mathbb{P}^d$ nor $V^d$, then there exists a torus invariant subsurface $S\subset X$ of Picard number two such that

$$\mathrm{ch}_2(X)\cdot S\le 0.$$

The conjecture strengthens the paper's explicit results by asserting the existence of such a surface uniformly for every smooth toric Fano variety outside the two exceptional families; its general status is left open by the supplied text.