If I have a normally distributed variable $\mathcal{N}(\mu,\frac{1}{\tau})$ with fixed $\mu$ then the conjugate prior for an unknown $\tau$ is then $\mathcal{Ga}(\frac{n}{2}+\alpha, \beta + \sum_i \frac{(x_i - \mu)^2}{2})$.
If my precision is instead $\tau = \frac{1}{\sigma_a^2 + \sigma_b^2}$, where $\sigma_a^2$ is known and $\sigma_b^2$ is unknown, is there a form of the conjugate prior for $\sigma_a^2$?
It seems as though this should be simple but I have struggled with it. The application of this is to sample $\sigma_b^2$ in a Gibbs sampler.