The finite basis problem for the flat semirings $S(W)$

We focus on the finite basis problem for flat semirings of the form $S(W)$, where $W$ is an arbitrary set of nonempty words. We prove that $S(W)$ generates a Cross variety (and hence is finitely based) whenever every word in $W$ has length at most $3$, whereas it is nonfinitely based whenever there exists $k \geq 3$ such that $W$ is $x^{k+2}$-free but not $x^{k+1}$-free. In particular, if $W_k$ denotes the set of all words of length $k$, then $S(W_k)$ is finitely based if and only if $k \leq 3$. Moreover, $S(W)$ is nonfinitely based whenever $W$ is finite and not $x^4$-free. These results provide a partial answer to an open problem raised by Jackson et al.~(J Algebra 611: 211--245, 2022).