Kato's conjecture on the algebraic cycle representing the monodromy operator

Let $S^{[3]}$ and $S^{[4]}$ be the strata appearing in the semistable degeneration, and let $[N]$ denote the cohomology class of the monodromy operator. The relevant restriction map is the alternating sum of restriction maps from $S^{[3]}$ to $S^{[4]}$. Kato's conjecture. There exists an algebraic cycle on $S^{[3]}$ whose cycle class lies in the kernel of the restriction map to $S^{[4]}$ and which maps to the class $[N]$ of the monodromy operator discussed above. This is the consequence of the Hodge conjecture identified by Kazuya Kato in the setting of Lefschetz fibrations; the source does not state a resolution, so the conjecture is left open.