Kuca's regularity classification conjecture for Ulam and modified Ulam sequences

Let $a<b$ be relatively prime positive integers. Let $\mathcal{U}(a,b)$ denote the Ulam sequence generated by $a,b$, let $\mathcal{V}(a,b)$ denote the corresponding modified Ulam sequence, and call either sequence regular when it is eventually a finite union of arithmetic progressions. Kuca's regularity classification conjecture. Each of the two sequences is regular if and only if it has finitely many even terms: $\mathcal{U}(a,b)$ is regular if and only if $\mathcal{U}(a,b)$ has finitely many even terms, and $\mathcal{V}(a,b)$ is regular if and only if $\mathcal{V}(a,b)$ has finitely many even terms. Finch's theorem establishes the implication from finitely many even terms to regularity for $\mathcal{V}$-sequences; the converse, and the corresponding full assertion for Ulam sequences, are presented as open.