Conjecture on Verlinde coefficients and fusion rules for simple VOSA modules

Let $p,p'$ satisfy $p'\geq 2$, let $\lambda=(r,s;\theta)\in\mathscr{S}_{p,p'}$, let $\mu_j=(r_j,s_j)\in K_{p,p'}$, and let $x_j\in\mathbb{C}\setminus iS_{r_j,s_j}$ for $j\in\{1,2\}$. Define the atypical-typical-typical Verlinde coefficient by

$$N^{(\mu_2;x_2)}_{\lambda,(\mu_1;x_1)}:=\sum_{\mu'\in K_{p,p'}}\int_{\mathbb{R}}\mathrm{d}x'\frac{S_{\lambda,(\mu';x')}^{at,(0,0)}S_{(\mu_1;x_1),(\mu';x')}^{tt,(0,0)}S_{(\mu_2;-x_2),(\mu';x')}^{tt,(0,0)}}{S_{(1,0;0),(\mu';x')}^{at,(0,0)}}.$$

Verlinde-coefficient conjecture. The Verlinde coefficient $N^{\mu_2}_{\lambda,\mu_1}$ can be expressed as a certain delta function with a non-negative integer coefficient, and this integer coincides with the fusion rule of the corresponding simple $L_{c_{p,p'}}$-modules, namely the dimension of the space of intertwining operators of type

$$\left(\begin{array}{c}\mathcal{L}_{r_2,s_2;ix_2}\\ \mathcal{L}(r,s)^\theta\ \ \mathcal{L}_{r_1,s_1;ix_1}\end{array}\right).$$

This conjecture proposes that the analytically defined Verlinde coefficients reproduce the fusion multiplicities for the simple modules of the $\mathcal{N}=2$ superconformal algebra. The paper does not provide a resolution of the general assertion.