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Yes, this follows by the dominated convergence theorem, because for $\gamma\asymp1/n^2$ the product in the integrand is eventually (that is, for large enough $n$) no greater than $(1+\|t\|)^C$ for some real $C$, where $t=(t_1,\dots,t_n)$ and $\|t\|=\sqrt{\sum_1^n t_i^2}$, and $$\int_{\mathbb R^n}dt\,e^{-\|t\|^2/2}(1+\|t\|)^C<\infty. $$