Not if you are not gluten intolerant or celiac, it shouldn't. For those of us who are very sensitive, we get more sensitive over time to trace contamination, and sometimes this matters a lot, depending on how the manufacturing or packing facility handles the ingredients, do they have a dedicated, gluten free line, or do they run gluten or wheat products on the same lines, then wash them down afterwards ? I've reacted to some processed foods that are listed on the label as "gluten free" but at the same time I've sometimes switched brands to one with no gluten free labeling, but no gluten ingredients AND good manufacturing processes and dedicated lines, plus the warning says "may be processed in a facility that has traces of _______" (with no gluten listed) and had a successful outcome. These researchers who say the 20 ppm (that's 20 parts per million) is the amount of gluten cross contamination a regular celiac/gluten intolerant can take before showing symptoms, are dealing with what they believe to be the statistical averages, not the high and low ends of what really happens. Compare this to my spouse, who eats gluten free at home.... it doesn't matter the least to him, and he can switch back and forth from a "safe" menu for me, to a regular meal out. He just needs a bit more carbohydrate than I do, because we are different in how we process it.
Surprises in the brain, been there, done that, got the T- shirt.... you're in uncharted territory, but I hope it works out for you, whatever happens.