Counting word-representable digraphs with a prescribed number of strong components

Let an $\ell$-word be a word of length $\ell$ over an $n$-letter alphabet, and suppose it has a unique factorisation into factors corresponding to alphabet-disjoint partitions. A word-graph is the digraph represented by such a word, and a strong component is a maximal strongly connected subdigraph. Counting conjecture. The number of $\ell$-words over an $n$-alphabet, with a unique factorisation, that represent word-graphs with $k$ strong components is the number of $n$-partitions of an $\ell$-set with $k$ disjoint proper subsets of parts whose union is a set of the form ${j,j+1,\ldots,j+m}$. Each such union is an alphabet-disjoint partition in the corresponding representational word and is a strong component in the corresponding word-graph. This is presented as possible future work extending the paper's enumeration of strongly connected word-graphs; the text does not provide a proof or establish the asserted enumeration.