Cyclic insertion conjecture for multiple zeta values

For nonnegative integers $m_0,\ldots,m_{2n}$, define

$$Z(m_0,\ldots,m_{2n})=\zeta(\{2\}^{m_0},1,\{2\}^{m_1},3,\ldots,1,\{2\}^{m_{2n-1}},3,\{2\}^{m_{2n}}).$$

Cyclic insertion conjecture. For $n\geq0$ and $m_0,\ldots,m_{2n}\geq0$,

$$\sum_{j=0}^{2n}Z(m_{2n-j+1},\ldots,m_{2n},m_0,\ldots,m_{2n-j})=\zeta\left(\{2\}^{2n+\sum_{j=0}^{2n}m_j}\right).$$

This generalizes the Borwein–Bradley–Broadhurst Eulerian-value conjecture. The source attributes it to numerical work and gives no resolution.