Unicyclic-components conjecture for minimally $n$-convergent graphs

Let $\lambda_n$ denote the class of minimally $n$-convergent graphs, and let the components of a graph be its connected components. A component is unicyclic when it contains exactly one cycle.

Unicyclic-components conjecture. If $G\in\lambda_n$, then $G$ has unicyclic components.

The conjecture concerns the structure of minimally $n$-convergent graphs and is presented as a proposed structural description that would facilitate further results about these graphs.