A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results

For m\geq 2, let c_p(m) be the all-dimensional best constant in $ \left|\sum_{k=1}^m A_k\right|p \leq c_p(m)\left|\sum{k=1}^m |A_k|\right|_p. Tang and Zhang conjectured an explicit formula for every finite p>1. We disprove the conjecture with two explicit real 2\times 2 rank-one matrices at p=3/2. The comparison is certified by seven strict rational inequalities and, in particular, places the attained ratio above 207/200, while the conjectured constant lies below 207/200. On the positive side, we prove the conjectured sharp bound for every family of rank-at-most-one summands when 2\leq p<\infty, and classify all equality cases. We also prove the corresponding endpoint statement for p=\infty. Finally, for arbitrary complex matrices, we establish the conjectured sharp constant in the case m=2, p=4$.