A counterexample to the Shannon conjecture following Eq. (69) in Tarabunga et al., Section V.C

The Shannon conjecture following Eq. (69) in Section V.C of Tarabunga et al., arXiv:2411.05766v1, is false. An explicit three-qubit state has stabilizer-basis entropy log 4, whereas the proposed absolute-amplitude lower bound is log 16 - (9/8) log 3 > log 4. The same construction refutes the bound for nonnegative amplitudes, and continuity gives strictly positive examples. It extends to every N >= 3 and to a range of Renyi indices containing one. The collision entropy of the comparison distribution remains a valid Shannon lower bound for arbitrary affine supports. For amplitudes of constant modulus on each affine coset, the original bound holds at every Renyi index; this includes the source's Figure 1 product states.