This is a follow-up to this question.

Fix a finite first moment probability measure $q\in\mathcal{P}_1(\mathbb R ^d)$, and real numbers $K,M,R$. Consider the following set:

$$A:=\left\{p\in\mathcal{P}_1(\mathbb R ^d): \int |x|dp\leq K, \int x dp=M, \mathcal{W}_1(p,q)\leq R \right\}$$

Is it a $\sigma$-compact set in $(\mathcal{P}_1(\mathbb R ^d),\mathcal W _1)$? It is not hard to check that this set is closed and that it is tight (weakly). As was already seen in the original question, this set is not compact, but I am hoping that it is still $\sigma$-compact.

Many thanks in advance!