Why empirical likelihood ratio can be larger than some quantile of $\chi^2_1$, if EL is said to be maximised when $p_i = \frac{1}{n}$?

Question

If EL is defined as $\prod_{i=1}^n p_i$ and empirical likelihood ratio is defined as $R(F) = -2 \log \{L(\hat{F})/L(F_n)\}$. If $R(F)\ge c$, then we can reject the $H_0: \mu = \mu_0$. If EL is maximised when $p_i = \frac{1}{n}$, then why the ratio is ever larger than 1?