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Let $G$ be a group, and let $S$ be a finite subset of $G$ that generates $G$as a monoid. The co-word problem is the collection of words in the free monoid$S^{\ast}$ that represent non-trivial elements of $G$.  A current conjecture, based originally on a conjecture of Lehnert andmodified into its current form by Bleak, Matucci, and Neuh\"{o}ffer, says thatThompson's group $V$ is a universal group with context-free co-word problem. Inother words, it is conjectured that a group has a context-free co-word problemexactly if it is a finitely generated subgroup of $V$.  Hughes introduced the class $\mathcal{FSS}$ of groups that are determined byfinite similarity structures. An $\mathcal{FSS}$ group acts by localsimilarities on a compact ultrametric space. Thompson's group $V$ is arepresentative example, but there are many others.  We show that $\mathcal{FSS}$ groups have context-free co-word problem under aminimal additional hypothesis. As a result, we can specify a subfamily of$\mathcal{FSS}$ groups that are potential counterexamples to the conjecture.

Local similarity groups with context-free co-word problem