Bug spray for Baker/Rainier late June?

I'm doing a 10-day course with Alpine Ascents (6 days on Baker, 4 Days on Rainier) in a couple weeks - I leave June 17th.  They have bug spray on the gear list.  I've never been to the Cascades before - can I expect a lot of mosquitos this time of year?  I always hike with almost every patch of skin covered (hands included) regardless of temperatures - so it would only be my face exposed.  I hate bug spray with a passion, but if there will be lots of mosquitos I'll bring it.

I’ll be on Baker the same weekend. See ya out there!

bug nets work too and are very small/light

You are paying them a lot of money for this trip, call them and ask. I'm sure everything on their list has a reason for being on there.  

Personal experience, I never have needed bug protection above tree line on either of those mountains. When hanging out below tree line there were a decent number of mosquitoes.

None.  Biting and stinging insects will  be too few in number to have a concern.

You're not going to be hiking in (or out) on Baker with every inch of skin covered, you'll sweat your nuts off.  Especially for a 6 day seminar.  Your pack will be 60-80 lbs.

I was just there this weekend. Only a few mosquitos down low.