Let $\mu$ be a probability measure on $\mathbb{R}^d$ such that $S_\mu$ is its second moment matrix, means $$S_\mu=\int_{\mathbb{R}^d}xx^Td\mu(x)$$ I'm trying to prove the existence of $\mu^\epsilon$ a probability measure on $\mathbb{R}^d$ such that its second moment matrix $S_{\mu^\epsilon}=S_\mu+\epsilon I_d$ with ($\epsilon >0$) and $$W_2^2(\mu,\mu^\epsilon)\leq \epsilon$$ where $W_2^2(.,.)$ is the 2-Wasserstein distance