# Incorporating prior evidence of predictor having no effect in bayesian linear regression model

$$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon, \quad \epsilon \sim N(0, \sigma^2)$$ with the conjugate prior
\begin{align*} &\sigma^2 \sim \text{Inv-}\chi^2(\nu_n, \sigma_n^2)\\ &\beta | \sigma^2 \sim N(\mu_0, \sigma^2\Omega_0^{-1})\\ &\mu_0 = (0,0,0)\\ &\Omega_0 = 0.01 \cdot I_3\\ &\nu_0=1\\ &\sigma^2 = 10^4 \end{align*} Question: Now say that we learn that earlier research shows strong evidence that $$x_2$$ has no effect on $$y$$. How could we can change the model to account for this information? (There may be many different ways but I'm mostly looking for a standard way if there is such a thing).
How I have tried: My first thought was to change the prior in some way, but the prior mean $$\mu_0$$ is already $$0$$ so there's not much to do there. My second thought was then that the prior variance around the mean $$0$$ should decrease for $$\beta_2$$ since the earlier research increases our prior belief that the mean is $$0$$, so $$\Omega_0^{-1}$$ would have to be decreased in the entry corresponding to $$\beta_2$$, i.e. increasing the entry corresponding to $$\beta_2$$ in $$\Omega_0$$. Not sure if this is a reasonable approach. I was also thinking that setting up a hierarchical model would be suitable but I just started learning about that so not sure how I would do it.