Alternating Möbius function conjecture for the matching pattern poset

Let $P$ be the matching pattern poset of Dyck paths, and let $\mu$ denote its Möbius function. The rank of an interval is the difference between the semilengths of its endpoints.

Alternating Möbius function conjecture. The Möbius function is alternating, meaning that it is

$$\mu([x,y])\geq 0$$

on intervals of even rank and

$$\mu([x,y])\leq 0$$

on intervals of odd rank.

This conjecture is supported by computational evidence. It proposes a global sign rule for Möbius values according to interval rank, but no proof or resolution is given in the source.