Def on the ice axe. I would check conditions but one thing to think about is how much time you have. The cirque of the towers is quite a long hike in/out after a long drive down a dirt road, while the grand ( still a decent hike in) can be accessed more quickly in my opinion. I was facing the same decision having been to both and chose the Tetons for an August trip since I was In the winds more recently. That said, snow and ice may be your major factor.
Both are amazing, I liked the winds better for pure rock climbing but have only done the upper exum in the Tetons so take that with a grain of salt. Enjoy!
Right now the Tetons definitely need crampons and an axe (according to the NPS and Wyoming Whisky anyway). Not sure how much that'll change in the next ten days, but I'm planning on bringing that out there next week.
Wyoming Whisky has photos of the Upper Exum route conditions as recent as a few days ago, if that helps.
Definitely go to the Tetons. Both are worthy places, but the Winds have the potential to be miserable in early July with mosquitoes. I've never had that problem in the Tetons, thankfully.
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