TITLE: Computational complexity of deciding isomorphism of rational polyhedral cones
QUESTION [5 upvotes]: Let $C,C'$ be rational polyhedral cones in $\mathbb R^n$ both with non-empty interior. Rational means they are generated by vectors with rational entries. One says that $C,C'$ are isomorphic if there is a unimodular integer matrix $A\in Gl(n,\mathbb Z)$ such that $AC=C'$. 


Question. What is the computational complexity of deciding whether $C,C'$ are isomorphic?  I'm happy to consider either that the cones are given by finite generating sets (say with integer entries) or by defining inequalities (say over $\mathbb Z$).

REPLY [3 votes]: According to the abstract of "On the complexity of polytope isomorphism problems", 

[W]e derive that the problems to decide whether two polytopes, given either by vertex or by facet descriptions, are projectively or affinely isomorphic, are graph isomorphism hard. 

Skimming the proof of Theorem 3, I think it should also establish that your problem is GI-Hard.
This paper has 6 citations in MathSciNet, none of which appear to address this issue, so it is possible that this is the state of the art.