Assarf–Joswig–Paffenholz product decomposition conjecture for toric Fano manifolds

Let $X$ be a toric Fano $d$-fold with Picard number $\rho_X=2d-k$. Let $S_3$ be the del Pezzo surface obtained by blowing up $\mathbb P^2$ at three torus-invariant points.

Assarf–Joswig–Paffenholz conjecture. If $d>3k$, then $X$ is isomorphic to $(S_3)^m\times Y$, where $m$ is a positive integer and $Y$ is a toric Fano manifold of dimension at most $3k$.

Via the toric dictionary, this is equivalent to the corresponding splitting conjecture for smooth Fano polytopes. The paper proves a weaker statement with the bound $15k^2+37k+1$, so the asserted linear bound remains open.