DeBrota--Stacey, Section III, Conjecture 1 after Theorem 2: entropy saturation characterizes SIC measurements

We prove Section III, Conjecture 1 immediately after Theorem 2 (p. 5) in DeBrota and Stacey, Physical Review A 100, 062327 (2019): a Luders minimal informationally complete channel attains the Haar-averaged output-entropy bound if and only if its measurement is itself a SIC, in every positive finite dimension. An exact entropy-deficit identity gives quantitative bounds, finite two-design tests and an arbitrary-finite-POVM equality characterization by weighted projective two-designs. The same decomposition recovers the entropy-exchange theorem and explains why the postscript's spectral-average replacement requires its saturation hypothesis.