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A special case of this problem is the graph isomorphism problem. Interestingly, for this it is unknown whether it is solvable in polynomial time (relative to the number of vertices, so $n$ in your case), and also unknown whether or not it is $NP$-complete. As far as I know, Luks' algorithm is still state of the art (though I might be wrong), and that has runtime $O(2^{\sqrt{n \log(n)}})$.

Since this is a special case of your problem, its general worst case runtime will be unknown, too. Of course in this special case, one has only subsets of size 2 / simplices of dimension 1; as you point out, as soon as we allow arbitrary rank simplices, the above runtime cannot be achieved anymore, as the input alone has size $O(2^n)$.

EDIT: Actually, looking at the Wikipedia link I give, I discovered that there is a paper by Babai and Codenotti (2008), "Isomorhism of Hypergraphs of Low Rank in Moderately Exponential Time", where they give an algorithm that works in the same general time as Luks' algorithm for hypergraphs (and thus in particular simplical complexes) of bounded rank that has roughly the same general run time as Luks' algorithm for graph isomorphism. Of course that still does not answer the general question.

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A special case of this problem is the graph isomorphism problem. Interestingly, for this it is unknown whether it is solvable in polynomial time (relative to the number of vertices, so $n$ in your case), and also unknown whether or not it is $NP$-complete. As far as I know, Luks' algorithm is still state of the art (though I might be wrong), and that has runtime $O(2^{\sqrt{n \log(n)}})$.

Since this is a special case of your problem, its general worst case runtime will be unknown, too. Of course in this special case, one has only subsets of size 2 / simplices of dimension 1; as you point out, as soon as we allow arbitrary rank simplices, the above runtime cannot be achieved anymore, as the input alone has size $O(2^n)$.