Borwein–Bradley–Broadhurst Eulerian multiple zeta value conjecture

For a multi-index $\boldsymbol{k}=(k_1,\ldots,k_d)$, write $\zeta(\boldsymbol{k})$ for its multiple zeta value, and write $\{2\}^m$ for $m$ consecutive entries equal to $2$. For $m,n\geq0$, set $\mathrm{wt}=4(m+1)n+2m$. Borwein–Bradley–Broadhurst conjecture.

$$\zeta(\{\{2\}^{m},1,\{2\}^{m},3\}^{n},\{2\}^{m})=\frac{1}{2n+1}\frac{\pi^{\mathrm{wt}}}{(\mathrm{wt}+1)!}=\frac{1}{2n+1}\zeta(\{2\}^{\mathrm{wt}/2}).$$

These values were discovered numerically as conjecturally Eulerian multiple zeta values; the source reports extensive numerical evidence but no proof or resolution.