The parity-weighted double $T$-value conjecture

Let $T(p+i,q+j)$ denote a double multiple $T$-value, with $m\geq 1$, $p\geq 1$, and $q\geq 2$. Assume that $p+q+m$ is even. The parity-weighted double $T$-value conjecture.

$$\sum_{i+j=m \atop i,j\geq 0}\binom{p+i-1}{i}\binom{q+j-1}{j}T(p+i,q+j)\in\mathcal{Z}.$$

The conjecture is motivated by an explicit parity reduction for even-weight triple $T$-values and would provide further relations among double $T$-values. The source later notes that Murakami proved this conjecture except for its “only” aspects when discussing related conjectures.