Generation conjecture for multiple zeta values by symmetric Mordell–Tornheim values

Let $\Omega(\mathbf{k})$ be the real symmetric Mordell–Tornheim value associated with an index $\mathbf{k}\in\mathbb{N}^r$, defined by the limiting construction in the paper. Symmetric Mordell–Tornheim generation conjecture. The space $\mathcal{Z}$ is generated by the set

$$\{\Omega(\mathbf{k})\mid \mathbf{k}\text{ is an index}\}.$$

The paper reports that all multiple zeta values up to weight $12$ were written as $\mathbb{Q}$-linear combinations of such values. No general proof or resolution is supplied in the text.