Even if you increase protein, some of which will get converted into glucose…updated

What's Been in Kay's Brain? What's In Her Mind?: All of the essential amino acids have some degree of hydrophobicity…

…you could be creating a problem with weight gain as leucine, one of the BCAAs, and lysine are ketogenic versus glucogenic, meaning they do not get converted into glucose despite what some popular websites boast about the BCAAs and generally you will find leucine and lysine to be in equally high amounts in most animal proteins.   Leucine is also involved in the body’s production of cholesterol.

I found this information in Tom Brody’s Nutritional Biochemistry and Biochemistry, 2nd Edition, Moran, Scrimgeour, Horton, Ochs, and Rawn.