I have a question about bounding the Wasserstein loss between a continuous Gaussian mixture and a discrete uniform distribution of its centers. In particular, let $P=\frac 1 k \sum_{i=1}^k \mathcal{N}(\theta_i,\sigma^2 I_d) $ be a $d$-dimensional Gaussian mixture and let $Q=\frac 1 k \sum_i \delta_{\theta_i}$ be a uniform discrete distribution on the set of centers $\{\theta_1,\ldots,\theta_k\}$. For the case of $p=2$, my goal is to obtain the tightest bound as follows: $$ W_2(P,Q) \leq \text{some function of } (\sigma,d) $$
My attempt: Let $X=Z+\theta_D$ be a random variable where $Z \sim \mathcal{N}(0,\sigma^2 I_d)$ and $D\sim \mathsf{Unif}(\{1,\ldots,k\})$. Now define the random variable $Y=\theta_D$. Clearly $X \sim P$ and $Y\sim Q$. Then using the definition of $W_2$, we obtain that $$ W_2(P,Q) \leq \sqrt{\mathbb{E}\|X-Y\|^2} = \sqrt{\mathbb{E}\|Z\|^2}=\sigma \sqrt{d}. $$ So I am wondering if there are better bounds known for this specific case in the literature of optimal transport. Please point me to these references if it's already a well studied problem.