Let $G$ be a graph on the vertices $0, \dots, n-1$ and $k$ a number. Its closed-neighbourhood instance is the Hitting Set instance on the universe $\{0, \dots, n+1\}$ with the $n + 2$ sets $N[v] = \{v\} \cup N(v)$ for $v < n$, $\{n\}$ and $\{n+1\}$, and the solution size $k + 2$. A set of $k + 2$ elements meets all of them exactly when it contains $n$ and $n + 1$ and its other $k$ elements dominate $G$. So for $k \le n$ the graph has a dominating set of size $k$ exactly when the instance has a hitting set of size $k + 2$, and the solution size is at least $2$ and at most the universe, as the construction of lax-496464 needs.

The word of $(G, k)$ in lax-496464 is the compressed sparse row form of $G$ followed by $k$; it has $n + 1$ offsets, so $n$ is less than its length. A word RAM program reads it and writes the numbers of the closed-neighbourhood instance (in the sense of $JitFlowShopStandard$): for every vertex $v$ it tests each $u < n$ for membership in $N[v]$ by scanning the block of $v$, so the members come out in increasing order and without repetitions. When $k > n$, where there is no dominating set of size $k$, it writes instead the numbers of the instance with one empty set and solution size $2$, which has no hitting set. Then the word map of $JitFlowShopStandard$ writes the standard word of the zoo instance of the construction. Its machine count is the solution size plus one, $k + 3$ or $3$.

Theorem. $p$-Dominating-Set fpt-reduces to $FF(1,m) \mid\mid \sum Z_j$ parameterized by the number of machines $m$, by a polynomial-time map with $m \le k + 3$ whose words have numbers at most $64 (\ell + 1)^4$ for $\ell$ their length.

Since $p$-Dominating-Set is W[2]-complete (lax-496464), $FF(1,m) \mid\mid \sum Z_j$ is W[2]-hard for $m$, and strongly W[2]-hard: already on the words whose numbers are at most $64 (\ell + 1)^4$.

The reduction starts from Dominating Set and not from Hitting Set because the binary word of a Hitting Set instance writes the universe size in binary, so the universe can be exponentially larger than the word, while the construction has more than $n$ jobs.