Last I read the tunnel would be closed some time in early June -- okay, I see the text they use there is "on or around the beginning of June." Still, I would expect/hope that that means "no earlier than June 1st." which would put you in the clear.

That's what I've been hoping, anyway. My trip is a little dicier as I hope to pass through on June 2nd, I think. Fingers crossed. Let us know how you find the trail.