Vanishing containment probability conjecture for mesh patterns

Let $m$ be a mesh pattern with at least one shaded box, and let a random mesh pattern of length $n$ be chosen uniformly. Vanishing containment probability conjecture. The probability that the random mesh pattern of length $n$ contains $m$ tends to $0$ as $n$ tends to infinity. This conjecture contrasts with the classical permutation case, where the probability that a permutation of length $n$ contains any fixed permutation tends to $1$ by the Marcus–Tardos theorem; the preceding result shows that almost all sufficiently long mesh patterns have Möbius function zero on the interval from the empty pattern to the pattern.