Hexagon polyomino convex-hull area bound

Let $n$ congruent regular unit hexagons form an edge-to-edge connected system. Hexagon polyomino area bound. The area of its convex hull is at most

$$\frac{1}{6}\left\lfloor n^2+\frac{14}{3}n+1\right\rfloor.$$

The statement is presented in the supplied text as a bound or proposed higher-dimensional analogue, but no proof or resolution is given there. It concerns an analogue of the polyomino convex-hull maximization problem for regular polygons.