Fusion-rule conjecture for a family of rational lisse vertex algebras

For each positive integer $m$, let $r=3m-1$ and let $[i]$ for $i=0,1,\ldots,r$ denote the irreducible modules of a rational lisse vertex algebra of central charge

$$c=13-6\left(m+\frac{1}{m}\right).$$

Fusion-rule conjecture. There exists such a rational lisse vertex algebra whose irreducible modules satisfy

$$[i]\boxtimes[j]\cong\bigoplus_{\substack{|i-j|\leq k\leq\min\{i+j,r-i-j\}\\ k\equiv i+j\bmod 2}}[k].$$

The claim is proposed from explicit low-rank computations of fusion rings for type $D_n$; no proof or disproof is given in the supplied text.