Classification conjecture for 2-ary order 2 Conolly recursions

For $\beta>0$, consider 2-ary order $2$ $(\alpha,\beta)$-Conolly recurrences of the paper's general form. The notation $\{a,b\}$ in the listed parameter positions means that either value may be chosen, and the displayed numbers count the resulting recurrences. The three parameter pairs considered are $(-2,3)$, $(0,2)$, and $(2,1)$.

Classification conjecture. For $\beta>0$, the only 2-ary order $2$ $(\alpha,\beta)$-Conolly recurrences are exactly the recurrences listed in the source for $(\alpha,\beta)=(-2,3)$, $(0,2)$, and $(2,1)$, with respective listed counts $1,3,18,81$; $4,16,4,16$; and $1,4,8,8,16$.

This is a finite computational classification conjecture. The supplied text reports that the parameter searches were checked experimentally, but gives no proof or resolution.