Tateo's dilogarithm identities for RSG/SG Y-systems

Let $L(x)$ be the Rogers dilogarithm, let $\mathbb{Y}_{\mathrm{RSG/SG}}(n_1,\dots,n_F)$ be the RSG/SG $Y$-system, and let $\mathcal{I}_+:=\{(a,m,u)\mid Y^{(a)}_m(u)\in\mathcal{Y}_+\}$. For a real positive solution, with $r$ and the parameters $p_a,q_a$ as defined for the RSG/SG systems, the following identities are conjectured for the variables in $\mathcal{I}_+$:

Tateo's dilogarithm identities.

$$\frac{6}{\pi^2}\sum_{(a,m,u)\in\mathcal{I}_+\atop 0\leq u<2r}L\left(\frac{1}{1+Y^{(a)}_m(u)}\right)=M_+,$$ $$\frac{6}{\pi^2}\sum_{(a,m,u)\in\mathcal{I}_+\atop 0\leq u<2r}L\left(\frac{Y^{(a)}_m(u)}{1+Y^{(a)}_m(u)}\right)=M_-.$$

For the RSG case,

$$M_+=r\left(-6A_F+\sum_{a:\mathrm{even}}n_a+2\right),\qquad M_-=r\left(6A_F+\sum_{a:\mathrm{odd}}n_a-4\right),$$ $$A_F=\sum_{a=1}^{F-1}(-1)^{a+1}\frac{1}{p_aq_a}+(-1)^{F+1}\frac{1}{p_Fr}.$$

For the SG case,

$$M_+=r\left(\sum_{a:\mathrm{even}}n_a+1\right),\qquad M_-=r\left(\sum_{a:\mathrm{odd}}n_a\right),$$

with the summation in the SG formulas running over $1\leq a\leq F$. These identities are the second main result proved in the paper, so the conjecture is solved by the stated results. They give explicit Rogers-dilogarithm sums for positive solutions of the RSG/SG $Y$-systems, extending the identities conjectured by Tateo.