Second shuffle conjecture for symmetric elliptic Hodge correlators

Let $E$ be a CM elliptic curve, let $s$ be a base point, and let $D$ denote the depth filtration on the space of symmetric elliptic Hodge correlators. The second shuffle relations are the relations obtained by the appropriate shuffle combinations of these correlators. Second shuffle conjecture. The second shuffle relations for symmetric elliptic Hodge correlators hold modulo the depth filtration. The lower-depth terms are independent of the base point $s$. This would extend the known depth-two result, where second shuffle relations are equivalent to dihedral symmetry, to higher depth. Establishing these relations would allow the map relating Hodge correlators to Bianchi or modular complexes to descend in higher depth.