Algorithmic inverse decomposition-matrix conjecture for the principal block

Let $B_0$ be the principal block, let $\Lambda_\lambda$ and $\bar{\Gamma}_\lambda$ be the poset and graph subsets defined by the algorithm, and let $d(\mu,\lambda)$ be the grading distance. The notation $[L(\lambda):\Delta(\mu)](v)$ denotes the graded decomposition number. Inverse decomposition-matrix conjecture. The algorithm constructs the inverse of the $v$-decomposition matrix of $B_0$:

$$[L(\lambda):\Delta(\mu)](v)=(-1)^{d(\mu,\lambda)}v^{d(\mu,\lambda)}$$

if $\mu\in\bar{\Gamma}_\lambda$, and it is $0$ otherwise. This is conjectured for the standards in the minimal BGG resolution $\Delta_\bullet\to L(\lambda)\to0$ produced by the algorithm.