Do you mean to say that bsgs algo gives you 100% information if key is or is not in s given range k1-k2?

Or you say bsgs would find private key even if k1 k2 interval is incorrect?

I had a closer look at JLP bsgs code (not cuda you speak about) and it seems there is no limit to 125 or 128 bit for search interval? Is bsgs by nature searching whole 256 bit range?