Well... almost. I did beat Solitaire while playing HITW, and I had a very good chance of winning that HITW game as well, but the lobby decided to crash in the middle of my 1v1 against my last opponent! What are the odds?
You'd be right for arguing I technically didn't actually "beat" HITW, but after nearly five hours of playing both HITW and Solitaire, I'm happy I managed to get this close to my objective. I had some other close calls as well: I beat Solitaire and placed second in my HITW game once, and I placed first in HITW without getting to finish my game of Solitaire multiple times.
So yes, in case you were wondering: beating HITW while also beating Solitaire is definitely possible, provided the servers don't crash.