Nonexistence of one-vertex JSJ quotients of a free product

Let $F$ be the free group appearing in the free product $F*\langle x,y\rangle$. A quotient is fully residually $F$ if it has the residual property relative to $F$, and its JSJ decomposition has vertex groups. One-vertex JSJ conjecture. There are no fully residually $F$ quotients of $F*\langle x,y\rangle$ whose JSJ decomposition has only one vertex group. This conjecture asserts that every such quotient must have a JSJ decomposition with more than one vertex group; the source presents it as a conjecture without providing evidence of a resolution.