The classification conjecture for ten-vertex graphs with lazy cop number three

Let $G$ be a graph on 10 vertices, let $R_3=K_3\square K_3$ be the $3\times3$ Rooks graph, and let $c_L(G)$ denote its lazy cop number.

Ten-vertex classification conjecture. The only graphs $G$ on 10 vertices with $c_L(G)=3$ are either the Petersen graph, or a copy of $R_3$ with an additional vertex $v$ whose degree is at most 5 and whose adjacencies are chosen in a particular way so as not to reduce the lazy cop number to two.

The conjecture is intended to characterize the graphs needed for the paper's proposed analysis of graphs with maximum degree four. The source describes the required adjacency conditions only qualitatively, and reports no proof or resolution.