rays in $H^1(G; \mathbb R)$, we refer to projective classes of homomorphisms $G \to \mathbb R$ with discrete/cyclic image. Without defining the BNS-invariant, we shall state its most relevant property for our purposes: a rational ray in $H^1(G; \mathbb R)$ is [*symmetric*]{}, i.e., is in $- \mathcal C(G) \cap \mathcal C(G)$ if and only if the corresponding class of homomorphisms $[p] : G \to \mathbb R$ have finitely generated kernel $K$; in this case, $ K $ is free for cohomological reasons [@FH99; @Bie81; @St68], $G \cong K \rtimes_{\phi}  \mathbb Z$ for some free group automorphism $\phi: K \to K$, and the natural projection $K \rtimes_{\phi} \mathbb Z \to \mathbb Z$ is in the projective class $[p]$. Fix a symmetric rational ray $r_0$ in $\mathcal C(G)$, and let $\phi_0: K_0 \to K_0$ be the corresponding free group automorphism. The presentation complex for $K_0 \rtimes_{\phi_0} \mathbb Z$ has a natural semi-flow with respect to the [*stable direction*]{} $\mathbb Z_+$. Dowdall-Kapovich-Leininger show in [@DKL17b] that getting from $r_0$ to any symmetric rational ray in the same component of $\mathcal C(G)$ amounts to reparametrizing this semi-flow (hence the convexity of the component) and with a careful analysis of this semi-flow, they are able 