Internal-zero conjecture for the pattern $1(l+1)l\dots 2$

Let $q=1(l+1)l\dots 2$ with $l\ge3$, and let the frequency sequence for $q$ be the sequence of pattern frequencies indexed by $n$. A frequency sequence has internal zeros if it contains a zero term between two nonzero terms. Internal-zero conjecture. The frequency sequence for $q$ has internal zeros for all $n\ge l+1$. The conjecture arises from numerical evidence and the contrast between earlier results, and it predicts that the claimed nonzero-frequency property fails for this pattern when $l\ge3$.