I have actually HQ'd an item when I was fully under the cap, twice now. One was a Bone Ring +1 (exciting, I know, right?), and the other was a synth of Horn Arrows. I cannot provide proof of this, unfortunately. I didn't screenshot them, just went "hmm, that's kinda cool," and kept grinding.
But yes, it can happen.
Bijou, to me, wrote:
I was you once. Welcome, you freakin' nutjob!!! We will add your distinctiveness to our pwn.
Nothing good can come of this. <.<;