Zagier–Hoffman–Saha dimension and basis conjectures for multiple zeta values and multiple t-values

Let $(d_n)_{n\geqslant1}$ be the Padovan sequence defined by $d_1=0$, $d_2=1$, $d_3=1$ and $d_n=d_{n-2}+d_{n-3}$ for $n\geqslant4$, and let $(f_n)_{n\geqslant1}$ be the Fibonacci sequence with $f_1=f_2=1$ and $f_n=f_{n-1}+f_{n-2}$ for $n\geqslant3$. Let $\mathcal{Z}_n$ and $\mathcal{T}_n$ denote the $\mathbf{Q}$-spans of all multiple zeta values and multiple $t$-values of weight $n$, respectively. A multiple zeta value of index $(i_1,\ldots,i_k)$ has weight $i_1+\cdots+i_k$, and a multiple $t$-value is denoted by $t(i_1,\ldots,i_k)$. Zagier–Hoffman–Saha conjectures. The following statements hold: $\dim \mathcal{Z}_n=d_n$; $\dim \mathcal{T}_n=f_n$; the set

$$\{\zeta(i_1, i_2, \ldots, i_k)\mid i_1+i_2+\cdots+i_k=n,\ i_j\in\{2,3\}\}$$

is a $\mathbf{Q}$-basis of $\mathcal{Z}_n$; and, for $n\geqslant2$, the set

$$\{t(a_1+1,a_2,\ldots,a_r)\mid a_1+a_2+\cdots+a_r=n-1,\ a_i\in\{1,2\},\ 1\leqslant r\leqslant n-1\}$$

is a $\mathbf{Q}$-basis of $\mathcal{T}_n$. These conjectures concern the expected dimensions and explicit bases of the spaces of multiple zeta values and multiple $t$-values; the source presents them as conjectures attributed to Zagier, Hoffman, and Saha, with no resolution supplied.