A uniform Tits alternative for endomorphisms of the projective line

A recent article of J.P. Bell, K. Huang, W. Peng and T.J. Tucker establishes an analog of the Tits alternative for semigroups of endomorphisms of the projective line. The proof involves a ping-pong argument on arithmetic height functions. Extending this method, we obtain a uniform version of the same alternative. In particular, we show that semigroups of $\mathrm{End}(\mathbb{P}^{1})$ of exponential growth are of uniform exponential growth.