I got a 5 game restriction and dropped to 0 on June 28th, raised mine up to 2 on August 10th. I played maybe 5-8 games a day and got at least 1 honor is 90% of them and at least two in 65+ish% of them. Just under two months is still to date the fastest I've seen someone raise honor (essentially, I haven't heard of anyone raising it any faster than me).
The real question is will you be able to raise it up before the season ends and rewards start coming out? The answer: Maybe, you'd have to grind which in turn can be mentally exhausting which might make you more toxic and tired. It MIGHT be possible though.