$T=\sum\log(1+X_i)\sim$ Gamma$(n,\lambda)$, and $E[(n-1)/T]=\lambda$

Let $X_1,\dots,X_n$ be iid with pdf $f(x\mid\lambda)=\frac\lambda{(1+x)^{\lambda+1}}$, $x>0$. With $T=\sum\log(1+X_i)$, the UMVUE of $\lambda$ is: