Equality in the recursive lower bound for degree-2n-2 fundamental models

Prove that the number of fundamental models in the simplex Δ_n of degree 2n−2 equals 2(a_1a_{n−1}+a_2a_{n−2}+⋯+a_{n−1}a_1), where a_k denotes the number of fundamental models in Δ_k of degree 2k−1.

Background

The paper constructs a family of fundamental models of degree 2n−2 by composing pairs of sharp models and proves that the displayed expression is a lower bound for their number. The authors state that they have no argument establishing the matching upper bound; computational evidence motivates the conjecture that the lower bound is exact.