Non-biautomaticity conjecture for irreducible lattices in products with tree automorphism groups

Let $H$ be a semisimple real Lie group with trivial centre and no compact factors. Let $T$ be the automorphism group of a locally finite unimodular leafless tree, and suppose that $T$ is nondiscrete. Let $\Gamma$ be an irreducible, non-residually finite, uniform $(H\times T)$-lattice.

Non-biautomaticity conjecture. The group $\Gamma$ is not biautomatic.

This conjecture would vastly generalise the paper's results. It concerns the possible failure of biautomaticity for irreducible lattices in products of semisimple Lie groups and nondiscrete automorphism groups of trees; its status is not resolved in the supplied text.