Given finite sets $A,B,C$ of finite strings, does there exist some $x,y \in A$ and some $z$ such that $x.z \in B$ and $y.z \in C$ hold (where $x.z$ denotes the concatenation of $x$ and $z$)? That is, do $B$ and $C$ contain a pair of sequences with a common suffix such that the respective remaining prefixes are both contained in $A$?
For example, $A := \{a,b,cd\}$, $B := \{ae,ada\}$, $C := \{bd,cdda\}$ satisfy the above condition for $a, cd \in A$, $ada \in B$, and $cdda \in C$.
Does this problem have an established name?