Dimension conjecture for alternating Mordell--Tornheim zeta values

For each weight $w\geq 3$, let $\AMT_w$ be the relevant $\Q$-vector space of alternating Mordell--Tornheim zeta values. For every Lyndon word $\bfk\ne(1)$ on the odd numbers, ordered by $1<3<5<\cdots$, let $\zeta(\bfk;1,\dots,1,-1)$ denote the corresponding alternating multiple zeta value, let $\lambda(\bfk)$ be its multiplicity, and let $|\bfk|$ be its weight. Define

$$C_w:=\left\{\prod \zeta(\bfk;1,\dots,1,-1)(2\pi i)^{2n}:2n+\sum_{\bfk}\lambda(\bfk)|\bfk|=w,\ n\geq0\right\}.$$

The alternating Mordell--Tornheim dimension conjecture. For every $w\geq3$, the space $\AMT_w$ can be generated by $C_w$. The conjecture proposes an explicit family of alternating multiple-zeta-value products generating every alternating Mordell--Tornheim space; the paper reports the claim in the context of known low-weight computations and dimension bounds, but no general proof is given.