The vanishing intersection conjecture for right-closure and conjugation invariants

Let $\operatorname{rcl}$ denote the right-closure operator, and let $\operatorname{im}\operatorname{rcl}$ be its image, the space of right-closure invariants. Let $\mathsf{ConjInv}$ denote the conjugation invariants. Vanishing intersection conjecture. For every dimension $d$,

$$\operatorname{im}\operatorname{rcl}\cap\mathsf{ConjInv}=\{0\}.$$

The statement is proved in the source for $d=2$ and is then conjectured in all dimensions. It asserts that no nonzero invariant is simultaneously right-closure invariant and conjugation invariant.