Direct-summand conjecture for the non-standard level modules

Let $0<|m|\leq 2$. Let $U(m)$ be the maximal submodule from Theorem, let $V(m)=V^{\Lambda_m}$ when $U(m)=\{0\}$, and write $Y(m)_-$ and $Y(m)_+$ for the lowest- and highest-weight parts of the quotient $Y(m)={\mathcal F}ib(m)/V(m)$. Then ${\mathcal F}ib(m)$ is the level-$m$ module for the rank 2 hyperbolic subalgebra ${\mathcal F}ib$.

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

$${\mathcal F}ib(m)=V^{\Lambda_m}\oplus Y(m)_-\oplus Y(m)_+.$$

The preceding proposition proves complete reducibility of the quotient $Y(m)$, while the conjecture supplies the missing assertion that the maximal submodule vanishes and that the non-standard irreducible module splits off from ${\mathcal F}ib(m)$.