Eventual vanishing conjecture for rational approximants

Let $L$ be the linear functional in the preceding generalization, with integer parameters $\alpha$, $\beta_i$, and $\gamma_i$, and let $P_n/Q_n$ be the associated rational approximants. Eventual vanishing conjecture. If $\alpha$, $\beta_i$, and $\gamma_i$ are integers, then

$$L(1)=\lim_{n\rightarrow\infty}P_n/Q_n=0$$

if and only if $P_n=0$ for all sufficiently large $n$. This proposes an arithmetic characterization of when the limiting value vanishes, but the supplied text gives no resolution.