Intrinsic-energy formula conjecture for $B^{r,s}$

Let $B^{r,s}$ be a simple affine crystal with extremal vector $u(B^{r,s})$, intrinsic energy $D_{B^{r,s}}$, level $4\operatorname{level}(B^{r,s})4$, and crystal statistic $4\varphi(b)4$. Let $H=H_{B^{r,s},B^{r,s}}$ be the local energy function on $B^{r,s}\otimes B^{r,s}$. Hatayama et al.'s intrinsic-energy conjecture. There is a unique $b^\natural\in B^{r,s}$ such that

$$\varphi(b^\natural)=\operatorname{level}(B^{r,s})\Lambda_0,$$

and

$$D_{B^{r,s}}(b)=H(b\otimes b^\natural)-H(u(B^{r,s})\otimes b^\natural).$$

The formula would recover intrinsic energy from the local energy against the distinguished element $b^\natural$; it is recorded as a conjectural property of the proposed crystals and is not established generally in the source.