Growth rate of distinct spanning tree numbers in all simple graphs

Show that the cardinality |'(n)| of distinct spanning tree numbers attained by all simple graphs on n vertices satisfies |'(n)| = e^{Ω(n log n)}.

Background

Dropping planarity, let |'(n)| count distinct spanning tree numbers over all simple graphs on n vertices. The initial values of |'(n)| grow quickly, and known bounds such as Cayley’s formula suggest super-exponential counts of graphs. The conjecture posits a strong lower bound on the variety of spanning tree numbers attained in this broader family.