Does there exist Eulerian quadrangulations that are not 1- or 2-degenerate? I am looking for Eulerian planar quadrangulations that are not 1- or 2-degenerate, but I cannot seem to find such graphs.
Note: a graph is Eulerian if and only if every vertex has an even degree. Clearly there exists such quadrangulations, for example, $K_{2,6}$ where the points in the partite set of $2$-points are placed on both sides of a line of the set with $6$ points. However, the case of creating a planar quadrangulation with each vertex having an even degree, and the quadrangulation also being $\ge 3$-degenerate, this becomes a lot more complicated. Are there any existing examples of such?

Remove the blue vertices to find a subgraph (which is itself a quadrangulation!) that has minimum degree $3$.

