Rational-span conjecture for zeta values and symmetrized Mordell–Tornheim values

Let $\zeta(n)$ denote the Riemann zeta value at the integer $n$, and let $\overline{\zeta}_m$ denote the symmetrized Mordell–Tornheim zeta value of order $m$. Rational-span conjecture. Every zeta value $\zeta(n)$ for $n\geq 2$ can be written as a $\mathbb{Q}$-linear combination of $\overline{\zeta}_{m}$ values with $m\geq 2$. The displayed identities verify this relationship in the first several weights, while the asserted generalization to arbitrary $n$ remains open.