The layered extremal word conjecture

Let $\Pi$ be a set of layered patterns, and let $\delta(\Pi)$ denote their packing density and $\delta'(\Pi)$ the corresponding density when the alphabet size is fixed as in the paper. A string is layered if it is formed from layers, and a maximal $\Pi$-containing string is one attaining the maximum number of occurrences of patterns in $\Pi$.

Layered extremal word conjecture. If $\Pi$ is a set of layered patterns, then

$$\delta'(\Pi)=\delta(\Pi),$$

and among maximal $\Pi$-containing strings in $[k]^n$, there is one which is layered.

This conjecture asserts that restricting to layered strings does not reduce the packing density for sets of layered patterns, and that a layered extremal string always exists. The supplied context does not establish the claim or give a resolution status.