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on Zagier's conjecture on multiple zeta values

The statement

For each natural number $n$ , let the weight of a list $(s_1, \ldots, s_k)$ of positive integers be $s_1 + \cdots + s_k$ . An admissible index is either the empty list or a list $(s_1, \ldots, s_k)$ with $s_1 \geq 2$ and all $s_i > 0$ . The multiple zeta value $\zeta(s_1, \ldots, s_k)$ is defined by

\zeta(s_1, \ldots, s_k) = \sum_{n_1 > n_2 > \cdots > n_k > 0} \frac{1}{n_1^{s_1} n_2^{s_2} \cdots n_k^{s_k}},

with the convention $\zeta(\emptyset) = 1$ . Let $\mathcal{Z}_n$ denote the $\mathbb{Q}$ -vector subspace of $\mathbb{R}$ spanned by all multiple zeta values $\zeta(s_1, \ldots, s_k)$ of weight $n$ with admissible index. Let the Zagier dimension sequence $d_n$ be defined by $d_0 = 1$ , $d_1 = 0$ , $d_2 = 1$ , and $d_n = d_{n-2} + d_{n-3}$ for all $n \geq 3$ . Determine whether the $\mathbb{Q}$ -dimension of $\mathcal{Z}_n$ equals $d_n$ for every natural number $n$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.