The injectivity conjecture for the homology-cylinder maps $a_k$

For every $k\geq 1$, consider the homomorphism

$$a_k:\mathbb Z_2\otimes {\sf L}_k\to {\sf K}^{\sf Y}_{4k-3}$$

from the exact sequence associated with the $A_n$-filtration of homology cylinders, where ${\sf K}^{\sf Y}_{4k-3}$ is the kernel appearing in that sequence. Injectivity conjecture for $a_k$. The homomorphisms $a_k$ are injective for all $k\geq 1$ and consequently

$${\sf K}^{\sf Y}_{4k-3}\cong \mathbb Z_2\otimes {\sf L}'_{2k}.$$

This is the homology-cylinder analogue of the injectivity question for $\alpha_k$ and of the nontriviality question for higher-order Arf invariants; the source presents it as unresolved.