The change-of-basis coefficients between the dual canonical and MV bases

Let $L$ be a finite-dimensional simple module over the relevant KLR algebra, let $d_L$ be the corresponding vector in the dual canonical basis of ${\mathbb C}[N]$, and let $b_Z$ be the MV-basis vector corresponding to an MV cycle $Z$. For each $L$ and $Z$, let $n_{L,Z}$ be the non-negative integer defined by the multiplicity of $Z$ in the scheme-theoretic support of the associated graded module constructed from $L$. Change-of-basis conjecture. The change of basis from the dual canonical basis to the MV basis is given by the non-negative integers $n_{L,Z}$; equivalently, the expansion of each $d_L$ in the MV basis has coefficients $n_{L,Z}$. The preceding theorem establishes the corresponding equality after applying the bar-character map, providing evidence for this conjecture; its general validity is left open.