Zagier's conjecture on multiple zeta values

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

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 $n$.