Graphs of free groups with cyclic edge groups

Let $X$ be a compact, connected, irreducible graph of graphs with circular edge spaces.

Graphs of free groups with cyclic edge groups. Then $\rho_+(X)\leq 0$. Furthermore, $\sigma_+(X)<0$ unless $\pi_1(X)$ has a Baumslag--Solitar subgroup.

The source expects this conjecture to be relatively easy to prove and presents it as a warm-up for more difficult examples; no proof or resolution is supplied.