Does prior containment work?

[maxbiostat]

Grinsztajn et al. (2021) argue (pp 6217) that some loose bounds for R0 are [1, 10]. I sort of agree. But what I wanna know is this: if I write

Pr(R0 \in   [1, 10]  | w) >= alpha,

where w are the prior hyperparameters and alpha \in (0, 1) is a prior probability level,
(i) is it easy to set w to achieve a certain alpha? How does that look for each of {gamma, log-normal, half-normal} priors?
(ii) Does this work in the sense of guaranteeing non-degenerate prior predictives and posterior inferences?
One reason to believe (ii) is shaky is that for (certain) gamma priors and the half-normal priors, the induced prior on R0 is heavy-tailed. Thus it might still assign non-trivial mass to intervals in the vicinity of 100, say.