For an irreducible character χλ of the symmetric group Sn, let Nn−(λ) = #{µ ⊢ n : χλ(µ) < 0}, where conjugacy classes are counted uniformly. OPAC-038 asks whether, for all sufficiently large n, this quantity is maximized by the sign representation. If p(n) is the partition number and q(n) is the number of self-conjugate partitions, then the sign row has exactly (p(n) − q(n))/2 negative entries. We prove several direct results toward this extremal problem. First, we give exact parity and sign-twist formulations and an exact positive-semidefinite fusion filter whose nullity is the number of negative entries. Second, we prove the desired inequality in every degree for the standard representation (n − 1, 1) and for the two-row family (n − 2, 2) by explicit reversible injections from negative conjugacy classes into odd conjugacy classes. Third, for every fixed partition ν, we prove that the stable family (n − |ν|, ν) satisfies the much stronger asymptotic estimate Nn−/p(n) → 0. Thus any infinite counterexample family must leave every fixed-tail regime. The global problem remains open; the results isolate the genuinely difficult regime to partitions whose distance from the first row grows with n.