It's actually okay to do this so long as your keys are truly unrelated you never reuse keys!  I ... don't know why you would ever do this! but there is some madman that appears to be doing this on the Bitcoin blockchain.

(well, okay, I can give an example of why you might do this:  If I don't have to compute kG, I can do a signing operation in under 1 microsecond, or about 100x faster.  But its still darwin award grade stupid.)

The reason I brought it up is that it just to illustrate that the reduction argument wasn't complete. It's probably not possible to make it complete given the shortcomings for provable security for ECDSA in general, I wouldn't fault your paper for it; only to note that risk isn't _completely_ eliminated by the proof.