Reflection-symmetric diagonal synthesis: resolving a conjecture of Huang, Kosugi, Nishi and Matsushita

Combined with the upper bound of Huang, Kosugi, Nishi and Matsushita [J. Phys. Soc. Jpn. 93, 054002 (2024); arXiv:2310.06676v2], this resolves their Section 6 conjecture that 2^(2n-1)+2n-2 CNOTs are optimal in the worst case for exact ancilla-free CNOT/R_z synthesis of reflection-symmetric diagonal unitaries on 2n qubits. The explicit hard instance has phase pi/2 at the all-zero and all-one strings. Multiplicative characters force every nonzero even parity; birth-span dimension and return conservation give the linear term. The same substitution gives the unrestricted 2dn-wire count, but not depth optimality or radial-potential optimality.