Stronger Newell-Littlewood saturation conjecture

Let ${\sf Par}$ denote the set of partitions, let $c_{\alpha,\beta}^{\mu}$ be Littlewood–Richardson coefficients, and let $N_{\lambda,\mu,\nu}$ be Newell-Littlewood numbers. Assume the parity hypothesis

$$|\lambda|+|\mu|+|\nu|\equiv 0\pmod 2.$$

Stronger Newell-Littlewood saturation conjecture. If $N_{k\mu,k\nu,k\lambda}>0$, then there exist partitions $\alpha,\beta,\gamma$ such that

$$c_{k\alpha,k\beta}^{k\mu}c_{k\alpha,k\gamma}^{k\nu}c_{k\beta,k\gamma}^{k\lambda}>0.$$

The source states that this conjecture is equivalent to Newell-Littlewood saturation, via the Newell-Littlewood formula and Littlewood–Richardson saturation, but gives no general resolution.