Upper growth rates are realized by sum closed classes

Let an upper growth rate of a permutation class mean a real number that occurs as the upper exponential growth rate of that class, and let a sum closed class be a permutation class closed under direct sum. Upper growth-rate conjecture. Every upper growth rate of a permutation class is the growth rate of a sum closed class.

The conjecture would reduce the characterization of upper growth rates to sum closed classes, extending the paper's growth-rate methods. The source places it among the remaining steps toward a complete characterization of growth rates; the stated gap in the surrounding results is approximately $0.05176$.