Not the greatest solution, considering Pinning isn't documented anywhere and most Chrome users have no idea it exists. Which is a shame, as pinning is probably my favorite browser feature.
So, for me, +1 for pinned tabs != throttled.
The pinned tab for GMail has a blue dot for "attention needed". Though this is suboptimal because I think it represents both IMs and new mail.
Unpinning and restarting my browser brings them right back in pinned state, and it seems like nothing I do short of reinstalling my browser profile corrects this.
I've seen this happen on multiple machines, so I feel like it can't just be me.