Matching the $\chi^2$ degrees of freedom to the estimated mean

Let $X_1,X_2,\dots,X_n$ be a random sample from $N(\mu,\sigma^2)$, $\mu$ is unknown and $\sigma^2>0$. Let $q_{n,\alpha}$ be the $(1-\alpha)$th quantile of $\chi^2_{(n)}$ distribution. Consider the tests for testing $H_0:\sigma^2\le1$ Vs. $H_1:\sigma^2>1$: $T_1$: Reject $H_0$ if and only if $\sum_{i=1}^n(X_i-\bar X)^2> q_{n-1,\alpha}$ and $T_2$: Reject $H_0$ if and only if $\sum_{i=1}^n(X_i-\bar X)^2>q_{n,\alpha}$ then which of the following statements is correct?