The LinKaw*–Drop1 equality conjecture

Let $\mathsf{LinKaw}^*$ and $\mathsf{Drop1}$ be the two families of relations defined in the source, with

$$\mathsf{LinKaw}^*=\operatorname{span}_{\mathbb{Q}}\{(\phi(w_1\ast w_2)x)\ast w_3\mid w_1,w_2\in y\mathcal{H},\ w_3\in\mathcal{H}^{\geq 2}\}.$$

Here $\mathcal{H}$ is the source's word algebra and $\phi$, $\ast$, $x$, $y$, and $\mathsf{Drop1}$ have the definitions given there.

LinKaw–Drop1 conjecture.* Based on numerical experiments up to weight $17$,

$$\mathsf{LinKaw}^*=\mathsf{Drop1}.$$

This is presented as an equality between two families of relations; its resolution is not given in the supplied text.