Recursive formula for fundamental models of degree $2n-2$

For each $n$, let $\Delta_n$ denote the relevant $n$-state model space, and let $a_n$ be the number of fundamental models in $\Delta_n$ of degree $2n-1$. For $n\geq 3$, Proposition 1 gives the lower bound

$$2(a_1a_{n-1}+a_2a_{n-2}+\ldots+a_{n-1}a_1)$$

for the number of fundamental models in $\Delta_n$ of degree $2n-2$. Recursive-formula equality conjecture. In Proposition 1, equality holds; equivalently, the number of fundamental models in $\Delta_n$ of degree $2n-2$ is exactly

$$2(a_1a_{n-1}+a_2a_{n-2}+\ldots+a_{n-1}a_1).$$

The proposition establishes the displayed quantity as a lower bound, while the paper says that no argument is currently known proving it is also an upper bound.