Takemura's conjecture on the support of the Heun root-counting measure

Let $Q(z)$ have roots $a_1,a_2,a_3$, and let $\mu$ be the limiting root-counting measure from the Shapiro–Tater conjecture. For $i\in\{1,2,3\}$, let $\gamma_i$ be the curve consisting of all points $b$ satisfying

$$\int_{a_j}^{a_k}\sqrt{\frac{b-t}{(t-a_1)(t-a_2)(t-a_3)}}\,dt\in\mathbb R,$$

where $j$ and $k$ are the remaining two indices and the integration follows the straight interval from $a_j$ to $a_k$. Let $\Gamma_Q$ be the union of the three segments of these curves joining each $a_i$ to their common intersection point inside $Conv_Q$. Takemura's conjecture. The support of the limiting root-counting measure $\mu$ coincides with $\Gamma_Q$. This gives an explicit geometric description of the support whose existence and general three-segment shape are asserted in the Shapiro–Tater conjecture; its resolution is not supplied in the source.