Let $X\subset \mathbb{R}^d$ and $Y\subset \mathbb{R^k}$ be nonempty sets. Endow the product space $X\times Y$ with the product topology and let $\mathcal{B}_{X \times Y}$ be the corresponding Borel $\sigma$-field. Let $\mu$ be a probability measure on $(X\times Y, \mathcal{B}_{X \times Y})$ and define the projection maps $$\pi_X:X\times Y \mapsto X: (x,y)\mapsto x,$$ $$\pi_Y:X\times Y \mapsto X: (x,y)\mapsto y.$$
Are the following statements equivalent?
(A) The marginal probability measures $\mu_X$ on $X $ and $\mu_Y$ on $Y$ have compact supports.
(B) The support of $\mu$ is a compact subset of $X \times Y$.
The fact that (A) implies (B) is proved in the answer to this question:
Support of a probability measure vs support of its margins
On the other hand, I would say that a simple reasoning by contradiction allows to prove that $\text{supp}(\mu_X) \subset \pi_X(\text{supp}(\mu))$ and $\text{supp}(\mu_Y) \subset \pi_Y(\text{supp}(\mu))$. Thus, $\text{supp}(\mu_X)$ and $\text{supp}(\mu_Y)$ are closed (by definition of support) and bounded (as they're included in the coordinate projections of a compact set), therefore compact. Am I missing anything stupid?