A minimal qutrit counterexample to Conjecture 4.9 of Lesniewski and Ruskai

Lesniewski and Ruskai conjectured that the contraction coefficient of every monotone Riemannian metric under a unital stochastic map equals the Hilbert–Schmidt contraction on the traceless subspace. We disprove the conjecture with an explicit entanglement-breaking qutrit channel induced by a doubly stochastic $3\times 3$ matrix. A faithful diagonal state and a commuting traceless tangent give, simultaneously for every normalized monotone metric, the exact lower bound ${\eta }_{\kappa }^{\mathrm{R}\mathrm{i}\mathrm{e}\mathrm{m}}({\mathrm{\Phi }}_K)\ge 8896\mathrm{/}20007>(62+2\sqrt{61})\mathrm{/}225={\mathrm{\Lambda }}_2({\mathrm{\Phi }}_K^{†}{\mathrm{\Phi }}_K)$. The counterexample is entirely classical on a maximal abelian subalgebra. A theorem of Hiai and Ruskai establishes the conjectured identity for all unital qubit maps, so dimension three is minimal among full matrix algebras.