Finite-versus-infinite conjecture for Conolly meta-Fibonacci recursions

Fix an order $p>0$ and parameters $(\alpha,\beta)$. A Conolly meta-Fibonacci recursion of order $p$ is a recursion of the paper's general form whose solution is an $(\alpha,\beta)$-Conolly sequence.

Finite-versus-infinite conjecture. The number of order $p$ $(\alpha,\beta)$-Conolly meta-Fibonacci recursions is finite if $\beta>0$ and infinite if $\beta=0$.

The source describes this as a stronger result suggested by the experimental evidence. It notes that the infinite case is proved for $p=1$ and $p=2$, but the general fixed-order assertion is presented as a conjecture.