The dimension conjecture for level-one Demazure and Weyl modules

Let ${\mathfrak g}$ be the simple Lie algebra under consideration, let $\lambda$ be a dominant weight, and let $D(1,\lambda)$ and $W^1(\lambda)$ denote respectively the level-one Demazure module and the Weyl module introduced above. Dimension conjecture. Recently proved, one has

$$\dim D(1,\lambda)=\dim W^1(\lambda).$$

This identifies the dimensions of the level-one Demazure and Weyl modules for every dominant weight.