Tasaka's isomorphism conjecture for the depth-three kernel

Let $\mathbf{W}_{N,3}$ be the period-polynomial space defined in the paper, let $\mathbf{Vect}_{N,3}$ be the corresponding coefficient space, and let $E^{(2)}_{N,3}$ and $E_{N,3}$ be the matrices defined there. Let

$$\xi:\mathbf{W}_{N,3}\longrightarrow\mathbf{Vect}_{N,3},\qquad p\longmapsto\pi_1(p)E^{(2)}_{N,3},$$

which has image in $\operatorname{Ker}E_{N,3}$, and write $\widetilde{\xi}$ for the induced map

$$\widetilde{\xi}:\mathbf{W}_{N,3}\longrightarrow\operatorname{Ker}E_{N,3}.$$

Tasaka's isomorphism conjecture. The map $\widetilde{\xi}$ is an isomorphism. The conjecture concerns the depth-three case and is attributed to Tasaka. The paper does not report a resolution.