Odd Morita nontriviality conjecture for the classes $W_{2k}$

For each $k=2,3,4,\dots$, let $W_{2k}\in\mathsf{FG}_1^{2k\text{-loop}}(0)$ be the cycle described in the preceding lemma, which states that $W_{2k}$ is non-zero and satisfies $d_cW_{2k}=d_uW_{2k}=0$. Odd Morita conjecture. For each $k=2,3,4,\dots$, the element $W_{2k}\in\mathsf{FG}_1^{2k\text{-loop}}(0)$ represents a non-trivial homology class. This is presented as an odd analogue of the Morita-class conjecture; the supplied text gives no evidence of a resolution.