Broadhurst's algebra-basis conjecture for MZV(6)

Let $B^6=\{B^6_n,n\geq1\}$ be the Broadhurst set, where the sets $B^6_n$ are formed from iterated integrals indexed by Lyndon words in the letters $0$, $\xi_3^2$, and $\xi_2=-1$, subject to the stated exclusion of the subsequence $0\xi_2$. Let $\operatorname{MZV}(6)$ denote the algebra of multiple zeta values at sixth roots of unity. Broadhurst's conjecture. The set

$$B^6=\{B^6_n,n\geq1\}$$

is an algebra basis for $\operatorname{MZV}(6)$. The source notes that this was conjectured by D. Broadhurst in 2015 and that the paper proves the corresponding statement for motivic MZV(6), while the numerical algebra-basis claim remains conjectural here.