The posterior probability $P(H_0|Data)$ is guaranteed to be $0.04$.

The posterior probability depends on the choice of the prior distribution $\pi(\theta)$ and the likelihood ratio.

The posterior probability $P(H_0|Data)$ must be less than $0.04$ by the law of total probability.

The posterior probability $P(H_0|Data)$ is equal to the p-value times the prior odds.