The Wiener index conjecture for 5-connected maximal planar graphs

Let $G$ be a maximal $5$-connected planar graph with $n\geq 12$ vertices. The 5-connected Wiener index conjecture.

$$W(G)\leq \begin{cases} \frac{1}{30}n^3+\frac{3}{10}n^2-\frac{23}{15}n+32, & n\equiv 0\pmod 5,\\ \frac{1}{30}n^3+\frac{3}{10}n^2-\frac{23}{15}n+\frac{156}{5}, & n\equiv 1\pmod 5,\\ \frac{1}{30}n^3+\frac{3}{10}n^2-\frac{23}{15}n+\frac{168}{5}, & n\equiv 2\pmod 5,\\ \frac{1}{30}n^3+\frac{3}{10}n^2-\frac{23}{15}n+31, & n\equiv 3\pmod 5,\\ \frac{1}{30}n^3+\frac{3}{10}n^2-\frac{23}{15}n+\frac{161}{5}, & n\equiv 4\pmod 5. \end{cases}$$

This is presented as a conjectured sharp bound for $5$-connected maximal planar graphs, based on cited constructions and asymptotic results. Its resolution is not supplied here.