Nonvanishing Euler characteristic for Eulerian biprojection intervals

Let $\mathcal{P}$ be a subfactor planar algebra, with Euler characteristic $\chi(\mathcal{P})$ as defined above, and let $[e_1,\operatorname{id}]$ be its biprojection interval. An interval is Eulerian when it is graded and its Möbius function satisfies

$$\mu(a,b)=(-1)^{|b|-|a|}$$

for every $a\leq b$.

Eulerian-interval conjecture. If $[e_1,\operatorname{id}]$ is Eulerian, then $\varphi(\mathcal{P})$ is nonzero, and consequently $\mathcal{P}$ is w-cyclic.

The claim extends the Boolean-lattice nonvanishing conjecture because Boolean lattices are Eulerian and the relevant Möbius-function relation identifies the two Euler characteristics up to sign. It remains open in the stated generality.