Median of the mean/max of an iid sample of exponential variables

Question

I would like to know if one can obtain a simple analytic expression of the median of $T_n$ defined by $$ T_n = \frac{\overline X_n}{X_{(n)}}, $$ where $\overline X_n$ is the empirical mean and $X_{(n)}$ the maximum of an iid sample of exponential random variables. The value does not depend on the parameter so we can take an exponential of parameter $1$ WLOG. I calculated the density of $T_n$, it is $$ \frac{n!}{n^{n-1}} \ \frac{ I_{n-1}(n t-1)}{t^n} , \ t \in \Big[ \frac 1 n,1 \Big] $$ where $I_{n-1}$ is the density of the Irwin-Hall distribution (the sum of $n-1$ independent uniform rv on $[0,1]$), $$ I_{n-1}(t) = \frac n {2} \sum_{k=0}^{n-1} \text{sgn}(t-k) \frac{(-1)^k (t-k)^{n-2}}{k! (n-k)!}, \ t \in [0,n-1], $$ (sgn is the sign function) but I couldn't calculate the median from this. I'm sure there is a more clever thing to do, but I can't find it. Thanks for any help.