The extended half-space algorithm conjecture for NET maps

Let $f$ be a NET map, meaning a nearly Euclidean Thurston map. The extended half-space algorithm is an algorithm based on the extended half-space theorem for deciding whether $f$ is equivalent to a rational function.

Extended half-space algorithm conjecture. The extended half-space algorithm decides, in finite time, whether or not $f$ is equivalent to a rational function.

The algorithm is motivated by computations suggesting that it almost always determines rational equivalence in practice, but the finite-time termination assertion is not established in the supplied text.