I know that account had access to CS:S much earlier than Jun 13, 2014. The screenshot may not necessarily be fake though. Perhaps the game was family shared in months prior. What you implied with the screenshot is still very dishonest though. The claim that you were never on these servers appears to be dishonest as well. On multiple occasions throughout the year (some recent) you have connected to these servers using that account. It's hard for me to believe you weren't already aware of the existing ban for cheating.
I'm going to go ahead and reject this appeal.
Steam Wrote: 4:02 PM - George, of the jungle: was out
4:02 PM - George, of the jungle: bison, dude
4:02 PM - Brawl Bashin’ Bison: ???
4:02 PM - George, of the jungle: you're very rude towards alina
4:02 PM - George, of the jungle: how about unbanning her friend?
4:02 PM - George, of the jungle: I mean
4:02 PM - George, of the jungle: it's only gamebanana skins
4:02 PM - Brawl Bashin’ Bison: LOL
4:02 PM - George, of the jungle: ^^
4:02 PM - Brawl Bashin’ Bison: LOLOL
4:02 PM - George, of the jungle: lol