The 2–3 conjecture for a transcendence basis of multiple zeta values

Let $\zeta(s_1,\ldots,s_k)$ denote a multiple zeta value of weight $s_1+\cdots+s_k$. A word in the alphabet $\{2,3\}$ is called Lyndon when it is strictly smaller than each of its proper nontrivial suffixes in the chosen lexicographic order. The 2–3 conjecture. The set

$$\left\{\zeta(s_1,\ldots,s_k): k\ge1,\ s_j\in\{2,3\},\ \text{$s_1\cdots s_k$ is a Lyndon word}\right\}$$

gives a transcendence basis of the $\mathbb{Q}$-algebra of multiple zeta values. This conjecture would determine the transcendence degree in each weight and gives a concrete proposed basis indexed by Lyndon words on $\{2,3\}$. The supplied source gives no evidence of a resolution, so its status is left open.