The dP3-fibration maximum conjecture for reflexive toric Calabi–Yau cones

Let ${\cal X}$ be a reflexive toric Calabi–Yau $n$-fold with toric diagram $\Delta_{n-1}$, let $Y$ be its Sasaki–Einstein base, let $V(b_i^{*};Y)$ be the minimum volume, and let $\chi(\widetilde{X(\Delta_{n-1})})$ be the Euler number of the associated completely resolved toric variety. A dP$_3$ fibration is a toric Calabi–Yau whose relevant hyperplane slices are hexagons, as in the examples discussed in the source. dP$_3$-fibration maximum conjecture. The maximum of

$$V(b_i^{*};Y)\,\chi(\widetilde{X(\Delta_{n-1})})$$

for reflexive toric Calabi–Yau $n$-folds is attained by various, not necessarily unique, dP$_3$ fibrations. This is extrapolated from the cone over $\mathrm{dP}_3$ and two four-dimensional examples; the claim is speculative and remains open.