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It's a tricky one and I can see why you're asking the question. I don't know the answer from a chemical viewpoint. I assume it would need damp conditions and spores are designed to float in the air and spread so there must be some risk but moving it to a dry building it seems likely it will just be contaminating the air on a low level if untreated until eventually it dies. If treated it should be dead a lot quicker.
It seems more viable if there was no mould growing on it.
Thanks, yes important to be careful. The rug wasn't visibly mouldy nor did it have the orange mould on it, since I don't think it was in direct contact with it. It will be hoovered thoroughly with a hepa-filtered vacuum and I'll look into fungicidal treatment for it. My concern is for the fabric of the new building at this point and whether cross-contamination is a possibility.