Locally series-parallel graph characterization conjecture

Let $G$ be a possibly infinite graph and let $r\geqslant 3$ be an integer. A graph is $r$-locally series-parallel when each of its $r/2$-balls is series-parallel, and an $r$-local subdivision of a wheel or of $K_4$ is a subdivision $H$ of a wheel or $K_4$ contained in $B_H(v,r/2)$ for some vertex $v$ of $H$. Let $G_r$ denote the $r$-local cover of $G$.

Locally series-parallel graph characterization conjecture. The following are equivalent:

1. $G$ is $r$-locally series-parallel.
2. $G$ does not contain an $r$-local subdivision of a wheel or of a $K_4$.
3. $G_r$ is series-parallel.

This conjecture seeks a characterization of locally series-parallel graphs analogous to the corresponding characterizations for locally acyclic and locally chordal graphs. Series-parallel graphs are precisely the graphs of treewidth at most $2$, while the conjectured forbidden local subdivisions reflect the role of $K_4$ and wheels in this setting. The source provides no resolution, so the conjecture remains open.