Black to kill unconditionally  

A friend of mine has submitted me the following problem. It's supposed to be easy but there's a part we actually couldn't solve despite spending some time on it.

Black to kill unconditionally  

This was the solution given for the problem.

Black to kill unconditionally  

However, after this we didn't manage to kill White unconditionally anymore. As compared with the diagram above, the absence of the a-b exchange gives White more possibilities.

Unfortunately this variation wasn't described in the solution. In the end we think there's an issue with the problem but we're probably wrong :-) there must be something we haven't seen and I'd be very glad if someone pointed it out.

Best,

Timm