= Solution
For $0<\alpha,\beta<1$, both event probabilities tend to one, so the <posterior odds> tend to $\boxed{R}$. If $1-\beta>\alpha$, the event's <Bayes factor> decreases toward one. To see monotonicity, put $a=1-\alpha>\beta$ and write
$$
\frac{1-\beta^n}{1-a^n}=\frac{1-\beta}{1-a}\frac{\sum_{k=0}^{n-1}\beta^k}{\sum_{k=0}^{n-1}a^k}.
$$
The last quotient is a weighted average of the decreasing values $(\beta/a)^k$, with weights $a^k$. Adding the next, smaller value decreases the average.
Thus the selection rule becomes less informative as more teams try: eventually a positive study is almost certain even under the null. This explains the hotter-field interpretation in https://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.0020124[Ioannidis's model]. It describes evidence retained by this particular selection event, rather than proving that every individual study becomes less reliable or that the full accumulated evidence gets weaker.
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