Backdoor Designer Key Recovery in LowMC-M
This would be done by a last round attack. The attacker would collect a crib of known pairs of inputs that differ by the plaintext difference (01010001101011 in your example). They would then run through $2^{16}$ guesses for the last round key, for each guess they would be able calculate the difference between each pair in round $r-1$ (by running just one round of the decryption process) and look for instances of the target difference (0111101010100100 in your case). In cases where they had guessed incorrectly, the target difference would not show up (or perhaps once); in the case of a correct guess the target would show up repeatedly.
Once the last round key is known, this drastically cuts down the number of main keys that could have produced it and (depending on key schedule) the main keys that are possible can then be exhausted.