Claire Glanois's kernel-image conjecture for period-polynomial maps

Let $\mathbf W_{N,r}$ be the space of restricted even period polynomials, and let $\varphi^{(r)}_j$ be the maps on the polynomial spaces used in the preceding lemmas. For $r\geq3$, consider the image obtained by restricting the maps $\varphi^{(r)}_2,\ldots,\varphi^{(r)}_{r-2}$ to $\mathbf W_{N,r}$ and then applying $\varphi^{(r)}_{r-1}$. Claire Glanois's conjecture. For all $r\geq3$,

$$\operatorname{Im}\left(\varphi^{(r)}_{r-1}\circ\varphi^{(r)}_{r-2}\big|_{\mathbf W_{N,r}}\circ\cdots\circ\varphi^{(r)}_{2}\big|_{\mathbf W_{N,r}}\right)=\ker \varphi^{(r)}_{r}\cap\operatorname{Im}\left(\varphi^{(r)}_{r-1}\circ\cdots\circ\varphi^{(r)}_{2}\right).$$

This strengthens the preceding inclusion and is intended to describe the kernel of the final map in terms of restricted period polynomials. Its status is not resolved in the supplied text.