Extended Horn cone conjecture for Newell–Littlewood coefficients

Let ${\operatorname{Par}}_n$ be the set of partitions with at most $n$ parts, and let ${\operatorname{EH}}(n)$ be the extended Horn cone cut out by the extended Horn inequalities on $({\operatorname{Par}}_n^{\mathbb Q})^3$. Define

$$\overline{{\operatorname{EH}}}(n)={\operatorname{EH}}(n)\cap\{(\lambda,\mu,\nu)\in({\operatorname{Par}}_n)^3: |\lambda|+|\mu|+|\nu|\text{ is even}\}.$$

Extended Horn cone conjecture. If $(\lambda,\mu,\nu)\in\overline{{\operatorname{EH}}}(n)$, then

$$N_{\lambda,\mu,\nu}>0.$$

This conjecture asserts that the extended Horn inequalities, together with the parity condition, characterize positivity of Newell–Littlewood coefficients. The source gives the conjecture in the context of its proof of a theorem referring to the GOY conjecture, but provides no resolution here.