Whether you can extend the procedure that long does not really matter. The issue is whether you failed to comply with your obligations to your landlord as of the time that the notice was served, and that you then failed to comply with the notice. The landlord may agree to take your money on June 1, and you should certainly discuss that with him, but he has no obligation to do so.
Just what your landlord did to you is not clear. You should consult with a landlord-tenant attorney to determine whether you have a defense.