Mushrooms grown on oats, safe?

[Ihaveceliac 0]

Om Organic Mushroom Nutrition Lion's Mane Mushroom Superfood Daily Boost Capsule, 90 Count https://www.amazon.com/dp/B07NRFNR16/ref=cm_sw_r_cp_api_glc_fabc_KqQ5FbE9V92EP
 

Is something such as this product safe? On the bottle it claims that the mushrooms are grown on oats? I recently started my gluten-free diet and I am trying to keep oats and corn out of my diet. Is it possible to have a reaction to this if I am in fact, also sensitive to oats?
 

Also, where could I buy a good testing kit for gluten?

[Scott Adams 484]

I think it would depend on your sensitivity level to oats. If you are extremely sensitive then you probably should avoid this supplement, or find one where the mushrooms are not grown on oats. 

This reminds me of wheat grass, which was very popular (and may still be) to get in things like Jamba Juice. Pure wheat grass is gluten-free, but when you watched the people make it in front of you they sometimes got a seed or two that had been mixed in with the grass, before putting it in the juicer, thus, that batch, and the machine were contaminated with gluten. I suspect the same thing could happen when they process these mushrooms.