Maximal incompatibility conjecture for random balanced dichotomic PVMs

Let $g\geq 2$. For each $d$, let $P_1,\ldots,P_g$ be $g$ independent Haar-random projections of rank $d/2$ on $\mathbb C^d$, and let $A_1,\ldots,A_g$ be their associated dichotomic observables. The maximal incompatibility conjecture states that, in probability,

$$\lim_{d\to\infty}\tau(A_1,\ldots,A_g)=\frac{1}{\sqrt{g}}.$$

This would show that independent Haar-random balanced dichotomic PVMs are asymptotically close to maximally incompatible for every $g\geq 2$, matching the known lower bound and the upper-bound analysis up to the conjectured exact value.