Kaneko–Tsumura's conjecture on unramified multiple T-values

Let $T(k_1,\dots,k_d)$ be the multiple $T$-value of weight $k_1+\cdots+k_d$ and depth $d$, and let $MZV$ denote the $Q$-span of multiple zeta values. A multiple $T$-value is unramified when it belongs to $MZV$; duals are understood in the sense of the multiple $T$-value duality.

Kaneko–Tsumura's conjecture. For even weights, the values $T(k,l,m)$ with $k,r$ odd and at least $3$ and $q$ even, together with their duals, are in $MZV$. Together with the single $T$-values, these values and their duals are the only unramified multiple $T$-values. For odd weights, the values $T(p,1,r)$ with $p,r$ even, together with their duals, are in $MZV$. Together with the single and double $T$-values, these values and their duals are the only unramified multiple $T$-values.

This conjecture proposes a complete classification of unramified multiple $T$-values by weight and depth. The paper studies its motivic analogue and establishes necessity of the proposed conditions at least in depth less than four, while the full classification remains open.