The longer you're ill with an infection before getting an antibiotic, the longer it will take you to recover. True or not?

My husband has a sinus infection. It was approximately eight days from when he started to feel sick to when he got an antibiotic. That was Friday. It’s now Sunday and he’s lamenting that he’s not feeling all that much better. He says that he normally feels more like himself after two days of antibiotic. He seems somewhat better to me, though not 100%.

I contend that had he gotten the antibiotic sooner, he would have recovered more quickly. I’m not upset with him for not going to the doctor sooner; it really seemed like a bad cold based on his symptoms.

Any thoughts?