Serre-pair decomposition and bijectivity problem

Assume $K$ is algebraically closed, $q\in\mathbb K$ is not a root of unity, $V$ is a finite-dimensional nonzero $K$-vector space, and $R,L:V\to V$ are nilpotent linear maps satisfying the two Serre relations. Assume further that $V$ has no common nonzero proper invariant subspace for $R$ and $L$. Serre-pair decomposition conjecture. There exists a nonnegative integer $d$ and nonzero subspaces $U_0,\ldots,U_d$ such that

$$V=U_0\oplus U_1\oplus\cdots\oplus U_d,$$

with $RU_i\subseteq U_{i+1}$ for $0\leq i<d$, $RU_d=0$, $LU_i\subseteq U_{i-1}$ for $0<i\leq d$, and $LU_0=0$. Moreover, assuming this decomposition exists, whenever $0\leq i\leq j\leq d$ and $i+j=d$, the restrictions $R^{j-i}:U_i\to U_j$ and $L^{j-i}:U_j\to U_i$ are bijections. If these assertions do not follow from the assumptions, the source asks for a counterexample. The supplied text gives no resolution.