I am wondering whether the following result is true:

Let $\mathcal W_p(\mathbb R^d)$ be the Wasserstein space of order $p$ and let $\eta$ and $\gamma$ be two probability measures in $\mathcal W_p(\mathbb R^d)$, such that supp($\eta$) $\subset$ supp($\gamma$). Then there is a sequence of probability measures $(\eta_n)$ that converges to $\eta$ in $\mathcal W_p(\mathbb R^d)$ and such that $\eta_n$ is absolutely continuous w.r.t. $\gamma$, with $\frac{d\eta_n}{d\gamma}$ being bounded.

It seems to be a well-known result but I was not able to prove it nor to find a reference. Any help and suggestion are appreciated!