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 $(1+2\|t\|)^C$ for some real $C$, where $t=(t_1,\dots,t_n)$ and $\|t\|=\sqrt{\sum_1^n t_i^2}$, and $$\int_{\mathbb R^n}dt\,e^{-\|t\|^2/2}(1+\|t\|)^C<\infty. $$
Added: details on bounding the product: We have $|t_i-t_j|\le2\|t\|\le1+2\|t\|$ for all $i,j$, and we have $n(n-1)/2$ factors $|t_i-t_j|^{2\gamma}$ in the product. So, the product is no greater than $$(1+2\|t\|)^{2\gamma \,n(n-1)/2}\le(1+2\|t\|)^C$$ for some real $C$ and all large enough $n$, given that $\gamma\asymp1/n^2$.