Full-dimensionality conjecture for feasible regions of consecutive permutation patterns

Let $clP_k$ denote the feasible region of density vectors of consecutive permutation patterns of size at most $k$, and let $\mathcal{L}_k$ be the set of Lyndon permutations of size at most $k$. The preceding results place $clP_k$ inside an algebraic variety of dimension $|\mathcal{L}_k|$. Full-dimensionality conjecture. The feasible region $clP_k$ is full-dimensional inside a manifold of dimension $|\mathcal{L}_k|$. This conjecture asserts that the upper bound supplied by the Lyndon permutations is sharp; computations indicate that this holds for small values of $k$, while the general case remains open.