Bound on the largest partition part for Littlewood-Richardson Hilbert basis elements

Let $\lambda$, $\mu$, and $\nu$ be partitions with at most $r$ parts, and let ${\sf LR}_r^{\mathbb Z}$ be the semigroup of triples $(\lambda,\mu,\nu)$ with positive Littlewood-Richardson coefficient. Largest-part conjecture. If $(\lambda,\mu,\nu)$ belongs to the Hilbert basis of ${\sf LR}_r^{\mathbb Z}$, then

$$\nu_1\leq r.$$

The statement appeared in a previous preprint version of the work; the supplied text gives no proof or disproof, so its current status is unresolved here.