The Crew–Narayanan–Spirkl circular disproportionate division conjecture

Let $\mu_1,\dots,\mu_n$ be probability measures on the unit circle $S^1$, and let $\alpha_1,\dots,\alpha_n$ be non-negative reals with sum $1$.

Crew–Narayanan–Spirkl conjecture. There is a partition $[n]=P\cup Q$ into two nonempty sets and a partition $S^1=X\cup X^c$ into two intervals such that

$$\min_{i\in P}\mu_i(X)=\sum_{j\in P}\alpha_j, \qquad \min_{i\in Q}\mu_i(X^c)=\sum_{j\in Q}\alpha_j.$$

The conjecture would imply the lower bound $q=2n-2$ in the disproportionate division problem. No resolution is given in the supplied context.