Characterization of maximal elements of the $L$-positivity poset

Let $PC_n$ be the poset of unordered pairs $\{\alpha,\beta\}$ of compositions satisfying $|\alpha|+|\beta|=n$, ordered by

$$\{\alpha,\beta\}\leq\{\gamma,\delta\}\quad\Longleftrightarrow\quad L_\gamma L_\delta-L_\alpha L_\beta\text{ is }L\text{-nonnegative}.$$

For an unordered pair $\{\alpha,\beta\}$, define $\{\alpha\vee\beta,\alpha\wedge\beta\}$ by choosing the smallest permitted $m$ when $\beta$ can be found inside $\alpha$, and otherwise setting $\{\alpha\vee\beta,\alpha\wedge\beta\}=\{\alpha,\beta\}$. Maximal-element characterization. The maximal elements of $PC_n$ are exactly the pairs $\{\alpha,\beta\}$ satisfying

$$\{\alpha,\beta\}=\{\alpha\wedge\beta,\alpha\vee\beta\}.$$

This identifies the maximal elements of the poset governing $L$-positivity comparisons of products of fundamental quasisymmetric functions. The supplied text does not state whether the characterization has been proved or remains open.