A conditioning principle for Galton-Watson trees

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than $\text{eps}$, converges as $\text{eps}\downarrow 0$ in law to the regular $\mu$-ary tree, where $\mu$ is the essential minimum of the offspring distribution. This gives an example of entropic repulsion where the limit has no entropy.