The classification conjecture for multiple $T$-values in ordinary multiple zeta values

Let $T(p,q,r)$ denote a triple multiple $T$-value, let $T(k)$ denote a single $T$-value, and let $\mathcal{Z}$ be the space of ordinary multiple zeta values. By duality, restrict to $T$-values whose depth is at most half their weight. The multiple $T$-value classification conjecture. For even weights, apart from the single value $T(k)$, the only triple values in $\mathcal{Z}$ are $T(p,q,r)$ with $p,r$ odd and at least $3$, and $q$ even, together with their duals. For odd weights, apart from the single and double values, the only such values in $\mathcal{Z}$ are $T(p,1,r)$ with $p,r$ even, together with their duals. The claim is motivated by the parity theorem, which places all odd-weight double $T$-values in $\mathcal{Z}$ and reduces even-weight triple values to single and double values; the asserted “only” parts remain conjectural.