Measure-theoretic defect conjecture for Butson matrices

Let $H\in C_N(s)$, with $s\in\mathbb N$ minimal, and let $\mu$ be the probability measure associated to the random variable

$$\varphi(a,b)=\#\{(i,j)\mid a_i b_j H_{ij}=1\},$$

for uniformly distributed $(a,b)\in\mathbb Z_s^N\times\mathbb Z_s^N$. Write $\overline{\operatorname{supp}(\mu)}^{\,\operatorname{conv}}$ for the closed convex hull of the support of $\mu$.

Measure-theoretic defect conjecture. For $H\in C_N(s)$, with $s$ minimal,

$$d(H)\in\overline{\operatorname{supp}(\mu)}^{\,\operatorname{conv}}.$$

This is presented as a reformulation of the Butson defect sandwich conjecture, expressing the defect through the distribution of unit-entry counts under row and column multiplication. The source gives no resolution.