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We study lower bounds for the functional $\\mathcal{F}_{\\gamma}(\\mu) = {\\rm Ent}(\\mu) - \\frac{1}{2} W^2_2(\\mu, \\nu)$, where $\\mu \\in \\mathcal{P}_{\\gamma}, \\nu \\in \\mathcal{P}_{\\gamma}$, ${\\rm Ent}(\\mu) = \\int \\log\\bigl( \\frac{\\mu}{\\gamma}\\bigr) d \\mu$ is the relative Gaussian entropy, and $W_2$ is the quadratic Kantorovich distance. 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