# Total variation distance bounds on multivariate normals with different means and variances

I'm trying to find a way to obtain an upper bound on the total variation distance between two multivariate normal distributions, i.e.

$$\vert \vert \mathcal{N}(\theta_1, a I) - \mathcal{N}(\theta_2, b I) \vert \vert_{TV} \le k \cdot \vert \vert \theta_1 - \theta_2 \vert \vert$$ where $$\theta_1 \ne \theta_2$$ and $$a \ne b$$ are constants and $$k$$ is some constant. Alternatively, some other bound that results in a scalar times any normed distance of $$\theta_1$$ and $$\theta_2$$ works. The problem I'm running into is I can't seem to find many ways to bound total variation distances between multivariate normal distributions. There are plenty of toy examples where either both distributions have the same mean or both distributions have the same variance in which it's easy to get an exact value, and I've computed the exact total variation distance between these two distributions, but I need something cleaner. Any help would be greatly appreciated!