Flag conjecture for largest Littlewood–Richardson coefficients

Let $\lambda\vdash n$, and let $\mu$ and $\nu$ be partitions with $c^\lambda_{\mu,\nu}={\rm C}(n)$, where ${\rm C}(n)$ denotes the largest Littlewood–Richardson coefficient at size $n$. The flag conjecture. One of the two indexing partitions contains the other:

$$\mu\subseteq\nu\quad\text{or}\quad\nu\subseteq\mu.$$

The conjecture would describe the maximizers of the global largest Littlewood–Richardson coefficient. The source motivates it by computational evidence, and it remains open.