  We consider the log-Gamma polymer in the half-space with bulk weights
distributed as $\operatorname{Gamma}^{-1}(2\theta)$ and diagonal weights as
$\operatorname{Gamma}^{-1}(\alpha+\theta)$ for $\theta>0$ and $\alpha>-\theta$.
We show that in the bound phase, i.e., when $\alpha\in (-\theta,0)$, the
endpoint of the polymer lies within an $O(1)$ stochastic window of the
diagonal. This result gives the first rigorous proof of the pinned phenomena
for the half-space polymers in the bound phase conjectured by Kardar(1985). We
also show that the limiting quenched endpoint distribution of the polymer
around the diagonal is given by a random probability mass function proportional
to the exponential of a random walk with log-Gamma type increments.
