In Simple Proof of Security of the BB84 Quantum Key Distribution Protocol, the description of the entanglement-based version of BB84 indicates that Alice and Bob perform a Hadamard transform on their entangled qubits (steps 2 and 7 on page 2). I understand that an eavesdropper Eve may not know the exact place of the qubit on which the Hadamard transform occurs. However, at the same time, as noted here, "applying a Hadamard gate on each qubit of the Bell state results in no change -- the resulting state is again the Bell state." So if we are to assume that Eve has unconditional power, then we can imagine the case where Eve is somehow waiting until Alice makes the announcement about the location of qubits where she applied Hadamard before Eve makes her own measurements.
I can see how applying Hadamard is useful if Eve is making measurements before Alice is making her announcement; however, Eve can potentially wait. Are we assuming that this case is impossible? Maybe this is addressed in the paper, but I could not find that. Any help in clarifying my question would be appreciated.