Block shuffle relation for refined symmetric multiple zeta values

Let $\mathfrak{X}_{\mathrm{ev}}'$, $\mathfrak{X}_{\mathrm{ev}}^+$, $\mathfrak{X}_{\mathrm{od}}$, $\mathcal{H}$, and $L_B^\circ$ be the parity-graded spaces and block-regularization map defined in the source, and let $L_B$ be the corresponding open-path map. Let $\mu^2$ denote the scalar operation appearing in the source. Block shuffle conjecture. For $(u,v)$ in the indicated union,

$$L_B^\circ(u\mathbin{\raisebox{-.23em}{\SmallDiamondshape}}v)= \begin{cases} 0,&(u,v)\in\mathfrak{X}_{\mathrm{ev}}'\times\mathfrak{X}_{\mathrm{ev}}^+,\\ \mu^2L_B(u)L_B(v),&(u,v)\in\mathfrak{X}_{\mathrm{od}}\times\mathfrak{X}_{\mathrm{od}}. \end{cases}$$

This is based on numerical investigation and concerns a closed-path analogue of the block shuffle identity for refined symmetric multiple zeta values. Its status is unresolved in the source.