The Lyndon-permutation dimension conjecture for feasible regions

Let $P_k$ be the set of permutons, and let $clP_k$ denote the feasible region of pattern densities for permutations of size at most $k$. Let $\mathcal{L}_k$ be the set of Lyndon permutations of size at most $k$.

Lyndon-permutation dimension conjecture. The feasible region $clP_k$ contains an open ball of dimension $|\mathcal{L}_k|$.

The previously known lower bound came from the set of $\oplus$-indecomposable permutations, while an algebraic-variety argument gives the upper bound $|\mathcal{L}_k|$. The conjecture asserts that this upper bound is attained; the source notes that computations confirm it for small $k$.