Convergence conjecture for the auxiliary numerical-semigroup sum

For each positive integer $k$, let $\mathcal A_k$ be the family of sets used in the paper's construction, and let $\varphi = \frac{1 + \sqrt 5}{2}$. The relevant series is

$$\sum_{k = 1}^{\infty} \sum_{A \in \mathcal A_k} \varphi^{-\left\lvert(A + A)\cap[0, k]\right\rvert + \left\lvert A\right\rvert - k - 1}.$$

Convergence conjecture. The sum in the displayed series converges to a finite value.

The authors say this is supported by computed partial sums and heuristic arguments, but they do not have a proof.