5-Coloring reconfiguration of planar graphs with no short odd cycles

The coloring reconfiguration graph $C_k\left(G\right)$ has as its vertex set all the proper $k$-colorings of $G$, and two vertices in $C_k\left(G\right)$ are adjacent if their corresponding $k$-colorings differ on a single vertex. Cereceda conjectured that if an $n$-vertex graph $G$ is $d$-degenerate and $k\ge d+2$, then the diameter of $C_k\left(G\right)$ is $O\left(n^2\right)$. Bousquet and Heinrich proved that if $G$ is planar and bipartite, then the diameter of $C_5\left(G\right)$ is $O\left(n^2\right)$. (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when $G$ is planar and has no 3-cycles. As a partial solution to this problem, we show that the diameter of $C_5\left(G\right)$ is $O\left(n^2\right)$ for every planar graph $G$ with no 3-cycles and no 5-cycles.