Maximum Möbius value conjecture for rank-3 intervals in the matching pattern poset

Let $U$ and $D$ denote up and down steps, respectively, and let $[P,Q]$ be an interval in the matching pattern poset of Dyck paths. An interval has rank $3$ when its endpoints differ in semilength by $3$, and $\mu$ denotes the Möbius function of this poset.

Rank-3 maximum conjecture. The maximum absolute value of the Möbius function on intervals of rank $3$ is

$$(2n+1)\cdot n^2,$$

attained by the interval

$$[U(UD)^{n-1}D,U(UD)^{n+2}D].$$

This conjecture is supported by computations; it predicts the extremal Möbius value among rank-3 intervals but does not assert uniqueness of the attaining interval.