Unimodular regular triangulation conjecture for uniform clutters

Let $\mathcal C$ be a uniform clutter with incidence vectors $v_1,\ldots,v_q$. Its max-flow min-cut property means that the associated max-flow and min-cut optimization problems have equal integral optimum values. A regular triangulation is obtained from a lifting of the vectors, and it is unimodular when every simplex spans the same lattice as the full configuration. Unimodular triangulation conjecture. If $\mathcal C$ is a uniform clutter that satisfies the max-flow min-cut property, then the rational polyhedral cone

$$\mathbb{R}_+\{v_1,\ldots,v_q\}$$

has a unimodular regular triangulation. The claim links the max-flow min-cut property to the existence of a particularly well-behaved regular triangulation; no resolution status is supplied in the candidate context.