Harborth's edge-count conjecture for matchstick graphs

A matchstick graph is a graph drawn in the plane with each edge a straight-line segment of unit length, such that no two edges have a point in common unless that point is an endpoint of both edges. For each integer $n\geqslant 1$, consider matchstick graphs on $n$ vertices. Harborth's conjecture. The maximum number of edges in a matchstick graph on $n$ vertices is

$$3n-\lceil\!\sqrt{12n-3}\,\rceil.$$

This is an extremal question about plane unit-distance graphs. The conjecture was posed by Harborth; the supplied source does not state whether it has been resolved.