Okazaki–Smith conjecture on line defect half-indices for ${\rm SU}(3)_{-4}$

Let $\langle W_m\rangle_{\mathcal{N},1,0}^{{\rm SU}(3)_{-4}}$ denote the line defect half-index for the $m$-th Wilson line in the ${\rm SU}(3)_{-4}$ Chern–Simons theory with the indicated Neumann boundary condition. The quantities $\mathbb{II}_{\mathcal{N}}^{{\rm SU}(3)_{-4}}$, $\langle W_1\rangle^{{\rm SU}(3)_{-4}}$, and $\langle W_2\rangle^{{\rm SU}(3)_{-4}}$ are the previously defined basic half-indices. Okazaki–Smith conjecture. The general case of $\langle W_m\rangle_{\mathcal{N},1,0}^{{\rm SU}(3)_{-4}}$ is a product containing the factor

$$\left\{\begin{array}{ll} \mathbb{II}_{\mathcal{N}}^{{\rm SU}(3)_{-4}}, &m\equiv 0 \pmod{3}, \\ \langle W_1\rangle^{{\rm SU}(3)_{-4}}, &m\equiv 1\pmod{3}, \\ \langle W_2\rangle^{{\rm SU}(3)_{-4}}, &m\equiv 2\pmod{3}, \end{array}\right.$$

a factor $(1+q^{m/4}+q^{m/2})$ when $m\equiv 0\pmod{4}$ and $m\neq 0$, and a factor consisting of a single power of $q$ together with a sign $\pm1$. The conjecture proposes a uniform description of all these line defect half-indices from the three basic half-indices, with the additional factor specified only in the stated cases; determining the precise power of $q$ and sign requires the formulas in the surrounding theory.