The temperature during the moulding process will destroy any mould and yeast spores or cells, so that source can be eliminated straight away.
The first area that I would explore is whether the cause is you or your customer. Not easy, but I would certainly send your own samples to a lab for testing. Negative results may not prove that it was not your fault, but it will help your argument. Air sampling and equipment swabbing may also be useful.
Is it just one cusomer? A visit to the customer to see how they store and use your closures may identify a bad practice. Some bottlers concentrate on the food/drink and bottle and forget the closure's GMP.
Other areas to look at:
Mould is most commonly attributed to a moisture problem somewhere. You mentioned air conditioning. Is there a reservoir of mould within the air conditioning units themselves.
What was the condition of the cases when your customer took their samples if they were wet or damaged then this is probably the cause.
What is the packaging format that you supply the closures in. Is it a polybag in a cardboard case? If it is a polybag, is it a manual system or do you blow air in the bag to inflate the bag. If so check the air filters, they can hold spores and you then increase th problem by blowing spores into the bag.
If it is a woven bulk bag, how do you clean/wash/dry these bags?
Do the closures have a cardboard or similar insert? if there is dampness in this raw material, mould may develop.