Compatibility of the principal-subspace basis with the Leclerc–Thibon basis

Let $L_{\Lambda_0}$ be the level-one representation, let $W_0$ be its principal subspace, let $B_0$ be the canonical basis induced on $W_0$, and let $B^+$ be the Leclerc–Thibon canonical basis of $L_{\Lambda_0}$. Principal-subspace compatibility conjecture. One has

$$B_0\subset B^+.$$

Equivalently, the canonical basis $B^+$ is compatible with the principal subspace $W_0$. The claim is supported by direct calculations but is not proved in the supplied text.