Maximum 3-separator conjecture for planar quadrangulations

Let $n\geq 14$. A 3-connected planar quadrangulation is a 3-connected planar graph in which every face is bounded by a cycle of length four. Let $\mathcal{G}^\ast$ be the class constructed by starting with a 4-cycle, successively adding vertices as described in the source construction, and gluing two disjoint copies of the cube $Q_3$ along specified 4-cycles of the resulting quadrangulation. A 3-separator is non-trivial when it is a 3-separator in the sense used for these quadrangulations. Maximum 3-separator conjecture. The $n$-vertex 3-connected planar quadrangulations with the maximum number of 3-separators are exactly the $n$-vertex graphs in $\mathcal{G}^\ast$. The construction gives graphs with exactly $4n-14$ non-trivial 3-separators, within ten of the maximum possible value by the cited lemmas; determining whether these are the unique extremal graphs remains open.