Thakur's basis conjecture for positive-characteristic multiple zeta values

Let $q$ be the cardinality of the constant field, let $k$ be the coefficient field, and let $\zeta_A(\mathfrak{s})$ denote the positive-characteristic multiple zeta value indexed by $\mathfrak{s}$. For a positive integer $w$, let $\mathcal{I}^{\mathrm{T}}_w$ consist of all tuples $\mathfrak{s}=(s_1,\ldots,s_r)\in\mathbb{Z}_{>0}^r$, with $r$ varying, such that $\operatorname{wt}(\mathfrak{s})=w$, $s_i\leq q$ for $1\leq i\leq r-1$, and $s_r<q$. Define

$$\mathcal{B}^{\mathrm{T}}_w=\{\zeta_A(\mathfrak{s})\mid\mathfrak{s}\in\mathcal{I}^{\mathrm{T}}_w\}.$$

Thakur's basis conjecture. The set $\mathcal{B}^{\mathrm{T}}_w$ is a basis of the $k$-vector space $\mathcal{Z}_w$.

The conjecture was proposed by Thakur in analogy with Hoffman's basis conjecture. The paper's abstract states that this conjecture is proved, and that Todd's dimension conjecture follows as a consequence.