The one-parameter Böttcher--Wenzel conjecture for weighted partial permutation matrices

Let \(f(A,B;q)=Re[B,A],[B,A]_q\), where \([B,A]_q=BA-qAB\), and set \[c(q)=12(1+q+√2(1+q^2)).\] Chruściński, Kimura, Ohno and Singal conjectured that \(f(A,B;q)≤ c(q)\|A\|F²\|B\|F²\) for all complex square matrices and real \(q\), and proved it for \(2×2\) matrices and normal \(A\). We prove the conjecture in every dimension and for every real \(q\) whenever either \(A\) or \(B\) is unitarily similar to a matrix with at most one nonzero entry in each row and column. This class contains all normal and square-zero matrices, all weighted Jordan chains, and matrices with cyclic partial permutations. The constant \(c(q)\) is optimal within the class. The proof combines a Cauchy--Schwarz estimate with a canonical orthogonal decomposition on which the associated superoperator reduces to coefficientwise estimates or explicit positive semidefinite Jacobi matrices. For \(A^2=0\), we determine its exact largest eigenvalue and obtain the sharp bound \(f(A,B;q)≤ c(q)\|A\|ₒₚ²\|B\|F²\), an analogous result holds when \(B^2=0\). At \(q=0\), the corresponding relaxation-rate functional satisfies the optimal bound \(r(A,B)≤\|A\|ₒₚ²\|B\|F²\).