The principal-subspace basis conjecture for the Leclerc–Thibon basis

Let $B_0$ be the basis of the principal subspace $W_0$ obtained from the completed canonical basis, and let $B^+$ be the Leclerc–Thibon canonical basis of the level-one representation $L_{\Lambda_0}$. Let $B_{-\infty}$ be the compatible basis of $L_{\Lambda_0}$ obtained from the bases of the principal subspaces $W_n$.

Principal-subspace basis conjecture. The basis

$$B_{-\infty}$$

coincides with the Leclerc–Thibon basis

$$B^+.$$

The conjecture is motivated by the agreement of the finite-stage bases and by the expected compatibility of the completed canonical basis with principal subspaces; the general coincidence remains open.