Rational multiples of powers of pi among symmetrized Mordell–Tornheim zeta values

Let $\overline{\zeta}_n$ denote the symmetrized Mordell–Tornheim zeta value of order $n$, and let $n\geq 1$. A value is a rational multiplicity of $\pi^n$ if it has the form $r\pi^n$ for some $r\in\mathbb{Q}$. Rational-multiplicity conjecture. The only three cases for which $\overline{\zeta}_{n\geq 1}$ is a rational multiplicity of $\pi^n$ are $n=1$, $n=2$ and $n=4$, with coefficients $0$, $\frac{1}{3}$ and $\frac{19}{15}$, respectively. The claim concerns the exceptional exact evaluations among these symmetrized sums; whether any further rational multiples occur remains open.