Logarithmic upper bound for the minimal edges β(t) of planar graphs (not necessarily simple)

Prove that the minimal number β(t) of edges of a planar graph (allowing multiple edges) having exactly t spanning trees satisfies β(t) = O(log t).

Background

The function β(t) is the smallest number of edges of a planar graph (not necessarily simple) with exactly t spanning trees; multiple edges are permitted and β(t) < 3 α(t). It is known that β(t) = O(log t * log log t), which is close to the natural logarithmic bound that would follow from α(t) = O(log t). The conjecture seeks to remove the extra log log t factor.