Kirillov–Bazhanov–Reshetikhin dilogarithm identity for constant Y-systems

Let $X$ be a simply-laced connected Dynkin diagram of finite type with rank $r$, Coxeter number $h=h(X)$, and vertex set $I=1,\dots,r1$. Let $\ell\geq 2$ and let $y_m^{(a)}1\in\mathbb{R}_{>0}$ for $a\in I$ and $m=1,\dots,\ell-1$ be a positive real solution of the constant $Y$-system described in the source, with boundary values $y_0^{(a)}{}^{-1}=y_\ell^{(a)}{}^{-1}=0$. Here $h=r+1$ for $A_r$, $h=2r-2$ for $D_r$, and $h=12,18,30$ for $E_6,E_7,E_8$, respectively. Kirillov–Bazhanov–Reshetikhin conjecture. For these values, one has

$$\sum_{a=1}^r\sum_{m=1}^{\ell-1}\tilde{L}(y_m^{(a)})=\frac{r(\ell-1)h}{h+\ell}\frac{\pi^2}{6}.$$

The identity arose from the thermodynamic analysis of restricted solid-on-solid models and predicts a uniform dilogarithm evaluation for all simply-laced finite Dynkin types and levels. The source presents it as a conjecture of Kirillov, Bazhanov, and Reshetikhin; its resolution is not specified in the supplied text.