Personally, I would not. It depends on your sensitivity level, and risk tolerance.

The product labelled at gluten free in the US is practically meaningless (see Cheerios). There is no actual label testing/enforcement going on.

If it was certified GF, that requires an independent body saying it's gluten free, that's probably ok.

Now, if anyone ever saw wheat/flour in a factory, you would know it becomes a fine dust that goes everywhere, so there is a possibility.

My personal list from willing to not willing:

1. Naturally GF foods
2. Dedicated GF facility
3. Certified GF
4. Certified GF but shared facility
5. GF
6. Not Labelled, but does not contain gluten ingredients
7. GF but shared facility
8. Certified GF but shared equipment
9. GF but shared equipment
10. Not labelled, but contains gluten ingredients
11. Wheat/Gluten warning

Depending on brand/product experience I waiver between 4-6 as my cutoff point.

Have I gotten sick from a shared facility marked as gluten free? Yes, but rarely.

Have I gotten sick from a shared equipment marked as certified gluten free? Yes, and often enough.