Infinitude conjecture for two-generator Ulam sets

Let $G$ be a commutative group, and let an Ulam set in $G$ be generated from two initial generators using unique sums of two earlier distinct elements. Two-generator infinitude conjecture. Every Ulam set in a commutative group with two generators is infinite. Finite examples are known with more than two generators, while the source reports no finite two-generator example and leaves this claim unresolved.