Alternating odd-position conjecture for block decompositions

Let $B=(0;\ell_1,\ldots,\ell_n)$ be any non-trivial block decomposition of even weight $N$. Alternate odd positions conjecture.

$$\operatorname{Alt}_{\{\ell_i\mid i\text{ odd}\}}I_{\mathrm{bl}}^{\mathfrak m}(\ell_1,\ldots,\ell_n)=0.$$

The conjecture extends the analogous alternating identity for 123-multiple zeta values to arbitrary block lengths. It is proposed after the special case suggested by Borwein, Bradley and Broadhurst; no proof or disproof is given in the source.