Finite basis conjecture for relational structures without finite monomorphic decomposition

Let $\mu$ be a finite relational signature, and let $\mathscr S_{\mu}$ be the class of all relational structures of signature $\mu$ without any finite monomorphic decomposition. A structure $A$ embeds a structure $B$ if there is an embedding from $B$ into $A$. Finite basis conjecture. The class $\mathscr S_{\mu}$ contains a finite subset $\mathfrak A$ of pairwise incomparable structures such that every member of $\mathscr S_{\mu}$ embeds some member of $\mathfrak A$. This conjecture asks for a finite basis, under embeddability, for the structures without finite monomorphic decomposition; the source states that the preceding reduction result is known, but that the relevant theorem is not known to extend in general. The status of this finite-basis assertion is not established in the supplied text.