A three-block symmetry conjecture for multiple zeta values

Let $a_1,a_2,a_3,b_1,b_2$ be non-negative integers, and let $Z$ denote the multiple zeta values defined by

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

Three-block symmetry conjecture. For any non-negative integers $a_1,a_2,a_3,b_1,b_2$,

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

This is presented as a family of identities outside the cyclic scheme, supported by extensive numerical evidence. Its general validity remains open.