Complexity of Acyclic Hypergraph Isomorphism It is well known that the graph isomorphism problem restricted on trees is much easier than the general case. It can be done in logarithmic space (Jenner B, Lange KJ, McKenzie P. "Tree isomorphism and some other complete problems for deterministic logspace." DIRO, Université de Montréal. 1997)

I would like to know whether something similar is known for the hypergraph isomorphism problem. In that case instead of trees we would have alpha-acyclic hypergraphs (see Characterization 15 on page 20 of https://arxiv.org/abs/1403.7076 for the definition of alpha acyclicity). 

So, I am asking whether some non-trivial complexity bounds for the following problem are known: 
Given two alpha acyclic hypergraphs G and H, is there an isomorphism between G and H?

Of course if something is known for other forms of acyclicity (e.g. beta or gamma) that would be interesting too.
To complete the answer by Holf, it is claimed here DAM 145(3) that isomorphism is GI-complete in $\beta$-acyclic hypergraphs.