MSOL recognizability conjecture for 3-connected and general $k$-outerplanar graphs

Fix $k$ and consider $k$-outerplanar graphs, with the subclass of 3-connected $k$-outerplanar graphs. A graph property is recognizable if it is recognized by a finite-state tree automaton; it is MSOL-definable or CMSOL-definable if it can be expressed in monadic second-order logic or counting monadic second-order logic, respectively. The $k$-outerplanar recognizability conjecture. (i) For 3-connected $k$-outerplanar graphs, recognizability equals MSOL-definability. (ii) For $k$-outerplanar graphs, recognizability equals CMSOL-definability. The paper establishes the corresponding results for bounded-degree $k$-outerplanar graphs and proposes these statements as further cases of Courcelle's conjecture; the conjecture remains open in the stated generality.