Existence conjecture for the monodromy cycle

Let $s$ be the standard log point over a finite field, let $X$ be a projective vertical log smooth fs log scheme over $s$, let $d=\dim X$, and let $q$ be a section of the log structure not belonging to $k^\times$. Let $K(X\times X,2d,d-1)$ be the group from the finer logarithmic Tate conjecture, and let $N_q$ denote the monodromy operators on log étale cohomology. Monodromy-cycle conjecture. There exists a unique element of $K(X\times X,2d,d-1)$ inducing

$$N_q:H^m(X)_{\ell}\longrightarrow H^m(X)_{\ell}(-1)$$

for every $m$ and every $\ell\neq p$, and inducing the analogous operator on log crystalline cohomology. This expected cycle would realize monodromy motivically; the source presents it as suggested by the finer log Tate conjecture and gives no resolution.