Conjecture for the genus-zero free energy coefficient of multi-cut Toeplitz determinants

The large-$n$ expansion of the Toeplitz determinant is organized by the remainder $m$ of dividing $n$ by $2r+1$, and let $F^{\{0\}}$ denote the corresponding $O(1)$ coefficient as a function of the filling-fraction parameter $\epsilon$. For $m=j$ and $m=2r-1-j$, with $1\leq j\leq r$, the coefficient is the same; write it as $F^{\{0\}}(j)$. Conjecture for $F^{\{0\}}$. For $m=0$,

$$F^{\{0\}}=-\frac{2r+1}{4}\ln\left(\cos\left(\frac{\pi\epsilon}{2}\right)\right)+C_{0},$$

where $C_{0}$ is constant. For $1\leq j\leq r$,

$$F^{\{0\}}(j)=A_{j}\ln\left(\cos\left(\frac{\pi\epsilon}{2}\right)\right)+B_{j}\ln\left(\tan\left(\frac{\pi\epsilon}{2}\right)\right)+C_j,$$

where $(A_j,B_j,C_j)$ are non-zero constants depending on the remainder. The constants $(C_j)_{1\leq j\leq r}$ and $C_0$ are the same, and the constants $(A_j,B_j,C_j)_{1\leq j\leq r}$ and $C_0$ may be rational numbers. These conjectural formulas extend the one-cut case $r=0$ and concern the difficult computation of derivatives with respect to arbitrary filling fractions; proving them remains open because the required spectral curve is not known explicitly in that generality.