Albert–Atkinson conjecture on layered patterns maximizing avoidance

Let $S_n(q)$ denote the number of permutations of length $n$ that avoid a pattern $q$. A pattern is layered if it consists of decreasing subsequences whose entries increase from one layer to the next. Let $q$ be a non-layered pattern of length $k$, and let $Q$ be a layered pattern of length $k$. Layered-pattern maximality conjecture. For every positive integer $n$,

$$S_n(q)\leq S_n(Q).$$

If true, bounds for layered patterns would give bounds for all patterns of the same length. Numerical evidence supports the conjecture, but the general assertion remains open.