Do you have experience with ice axes and snow climbing? If not, this would not be the place to learn. The fact that you are asking the question makes me think you might not have that kind of experience, and the only reason to take an ice axe is to self-arrest in case of a fall. And if you haven't trained and practiced that technique, don't take any trails where you will need it. It is not something that comes naturally, and failure means severe injury or death. Ropes, unless you are trained and experienced, only make it more dangerous, not less.

June is still pretty early in the mountains, and there will certainly be snow, maybe lots of it, at 10,000 feet. How much? we won't know until May, when the weather is finally wrapping up.

I'm not sure that is the advice you wanted...but it's the best that I can give!

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