The dihedral-tensor conjecture for representations in the Johnson cokernel

Let $H$ be the symplectic representation underlying the Johnson cokernel, let $D_{2s}$ be the dihedral group of order $2s$, and write $[H^{\langle s\rangle}]_{D_{2s}}$ for the corresponding dihedral coinvariants. Let $\mathsf C_s$ denote the degree-$s$ part of the Johnson cokernel, and call an irreducible representation in it a representation of size $s$. Dihedral-tensor conjecture. All representations of size $s$ in $\mathsf C_s$ are contained in

$$[H^{\langle s\rangle}]_{D_{2s}}.$$

The conjecture is motivated by explicit calculations for small values of $s$, where the dihedral coinvariants account for all representations observed in the corresponding graded pieces of the Johnson cokernel. Whether this containment holds in every size remains open.