Finch's even-term conjecture for two-generator Ulam sequences

Let $a,b$ be relatively prime positive integers, with one of $a,b$ even. For an Ulam sequence $\mathcal{U}(a,b)$, Finch's even-term conjecture. If $a$ is even, then $\mathcal{U}(a,b)$ has $2+a/2$ even terms, precisely $\{a,a+2b,\ldots,a+ab,(2a+4)b\}$; if $b$ is even, then it has $2+b/2$ even terms, precisely $\{b,b+2a,\ldots,b+ba,(2b+4)a\}$. Earlier results in the paper establish lower bounds and the initial families of even terms; the source presents Finch's assertion as an open conjecture.