The 132-avoiding principal-interval Möbius conjecture

Let $\tau$ be a permutation, and let $[1,\tau]$ be the principal interval below $\tau$ in the permutation pattern poset, where $1$ is the unique permutation of length one. Write $\mu(1,\tau)$ for the Möbius function of this interval. A permutation avoids a pattern when it contains no subsequence order-isomorphic to that pattern.

Principal-interval Möbius conjecture. If $\tau$ avoids the pattern $132$ (equivalently, $312$, $213$, or $231$), then

$$\mu(1,\tau)\in\{-1,0,1\}.$$

This is the principal-interval specialization of the broader 132-avoidance program for the Möbius function of the permutation pattern poset. The source presents it as one of two conjectures supported by data, and gives no resolution.