Generalized cyclic insertion conjecture for Z-values

Using

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

Generalized cyclic insertion conjecture. For $a_1,a_2,a_3,b_1,b_2\geq0$,

$$\begin{aligned} &Z(a_1,b_1,a_2,b_2,a_3)+Z(a_2,b_1,a_3,b_2,a_1)+Z(a_3,b_1,a_1,b_2,a_2)\\ ={}&Z(a_1,b_2,a_2,b_1,a_3)+Z(a_2,b_2,a_3,b_1,a_1)+Z(a_3,b_2,a_1,b_1,a_2). \end{aligned}$$

This is presented as a further conjectural relation in the $Z$-notation and is attributed to Borwein–Bradley–Broadhurst–Lisonek; the source gives no resolution.