Sean Eberhard commented on my posts on diagonal groups (see here and here). He is correct; there is an association scheme preserved by the full diagonal group with n factors in the socle; it is non-trivial if n > 2. The details take a few pages to write out but the basic idea is completely fine.
This shows that an AS-free group (one which does not preserve a non-trivial association scheme) must be either 2-homogeneous or almost simple. Clearly every 2-homogeneous group is AS-free; there are almost simple examples, but they are rather strange and no pattern has emerged thus far.
I hope a preprint will go on the arXiv sometime soon. But you read it first here (in Sean’s comments over the last few days).
Peter Cameron's Blog 
The paper is now on the arXiv: https://arxiv.org/abs/1905.06569 .
As a bonus, we construct an association scheme (a refinement of the above) from the Latin hypercube consisting of all n-tuples of elements of a group G with product equal to the identity. This was previously known only for abelian groups if n>2.