Exceptional motivic block-shuffle relations

Let $p_{a,b,c}$ and $q_{a,b,c}$ be the polynomials and let $r_{a,b,c}$ and $r'_{a,b,c}$ be the integers defined in the statement. Let $I_{\mathrm{bl}}^{\mathfrak{m}}$ denote the motivic block iterated integral. Exceptional block-shuffle conjecture. For every even integer $w\geq0$, the first displayed relation in the source is conjectured, and for every even integer $w\geq4$, the second displayed relation in the source is conjectured, with exactly the coefficient sums, parity conditions, Kronecker deltas, and polynomials specified there.

The relations are inferred from unusually short lattice-relation vectors in numerical data and are written using additional multiple zeta values; the source does not report a proof or disproof.