The Lyndon basis conjecture for multiple zeta values via multiple $t$-values

Let $\mathcal{Z}_k$ be the weight-$k$ space of ordinary multiple zeta values. For a sequence $(k_1,\ldots,k_r)$, call it Lyndon when every proper right subsequence is greater than the full sequence in lexicographical order. The multiple $t$-value basis conjecture. (1) A linear basis of $\mathcal{Z}_k$ is

$$\{t(2)^n t(k_1,\ldots,k_r)\mid n,r\geq 0,\ \forall k_i\text{ odd},\ k_i\geq 3,\ 2n+k_1+\cdots+k_r=k\}.$$

(2) An algebra basis of $\mathcal{Z}$ is given by $t(2)$ and the values $t(k_1,\ldots,k_r)$ with every $k_i$ odd and at least $3$, where $(k_1,\ldots,k_r)$ is Lyndon. The conjecture seeks linear and algebraic bases of ordinary multiple zeta values in terms of multiple $t$-values; the source records numerical evidence and notes related results of Murakami.