Tasaka's conjecture on multiple period-polynomial relations

Let $S_{N,r}$ be the index set used to define the coefficient space $\mathbb{P}_{N,r}$, let $\mathbf{W}_{N,r}\subseteq\mathbb{P}_{N,r}$ be the space of polynomials satisfying the stated period-polynomial relation, and let $E_{N,r}$ and $I_{N,r}$ be the matrices defined in the source. Write coefficient vectors as $(a_{\overrightarrow{m}})$. Tasaka's conjecture. The linear map

$$\eta:\pi(\mathbf{W}_{N,r})\longrightarrow \operatorname{Ker} E_{N,r}$$ $$a_{\overrightarrow{m}}\longmapsto (a_{\overrightarrow{m}})(E_{N,r}-I_{N,r})$$

is an isomorphism. This conjecture identifies the kernel of $E_{N,r}$ with the image of the higher-depth period-polynomial space under the operator $E_{N,r}-I_{N,r}$. The source notes a gap in Tasaka's proposed proof of injectivity and states that injectivity is proved for $r=3$ in the cited work.