The containment conjecture for shuffle and stuffle multiple $T$-value spaces

Let $\mathcal{Z}$ be the space of ordinary multiple zeta values, and let $\mathcal{T}^{\sh}$ and $\mathcal{T}^*$ be the shuffle and stuffle spaces of multiple $T$-values. The containment conjecture. Both $\mathcal{T}^{\sh}$ and $\mathcal{T}^*$ contain $\mathcal{Z}$ as a $\mathbb{Q}$-subalgebra. The conjecture would imply that both spaces are modules over $\mathcal{Z}$; the source remarks that numerical evidence suggests their intersection is strictly larger than $\mathcal{Z}$.