3 problems

Direct-summand conjecture. For $0<|m|\leq 2$, the maximal submodule $U(m)$ is $\{0\}$, $V(m)=V^{\Lambda_m}$ is a direct summand of ${\mathcal F}ib(m)$, and

Racah–Speiser recursion conjecture. For $m=\pm1,\pm2$, the weights of $P(V^{\Lambda_m}_-)$ follow

Complete reducibility conjecture. For each $m\in{\mathbb Z}$, ${\mathcal F}ib(m)$ completely reduces into a direct sum of one trivial module on level $0$, one non-standard module o…