A counterexample to the Nordhaus-Gaddum product conjecture for the k-th Laplacian eigenvalue (Chen, Guo, Li and Wang, Conjecture 20)

Conjecture 20 of Q. Chen, J.-M. Guo, W.-J. Li and Z. Wang (Electron. J. Combin. 32(4) (2025) #P4.37) asserts that mu_k(G) mu_k(Gbar) <= n(n - k) for the k-th largest Laplacian eigenvalues of every graph G on n vertices and its complement, for every k <= 3n/4. W. Jiang disproved it first (Zenodo, doi:10.5281/zenodo.22238691, record of 6 September 2026), with a 16-vertex counterexample at k = 7 and violations for infinitely many n; versions 1 and 2 of this record did not cite that work, which we found after publishing them. This record gives a smaller counterexample and different infinite families. The split graph of the Fano plane (points form a clique, lines an independent set, incidences as edges; n = 14, k = 6) gives mu_6(G) mu_6(Gbar) = 63 + 7 sqrt(57) > 112. More generally, the split graph of a symmetric 2-(v, kappa, lambda) design is a counterexample whenever v^2 - 4 v lambda + 4 kappa^2 > (v + 2 - kappa + lambda)^2; this holds for every finite projective plane and for every Hadamard 2-(4t-1, 2t-1, t-1) design with t >= 2 (t = 1 gives no counterexample). None of these graphs is in Jiang's construction. The note gives the statement as printed, both Laplacian characteristic polynomials, exact proofs, and the relation to Jiang's record and to Gibson (arXiv:2609.16049), which refutes only the equality clause at k = 3n/4. The archive contains the graphs (graph6 and adjacency lists), a standard-library Python script that verifies the counterexamples with exact integer and rational arithmetic, and a separate numpy check. Version 3 (2026-10-27) adds the citation of Jiang's record to NOTE.md and README.md; nothing else changed. Searches and verification were carried out with Claude (Anthropic) under the direction of Kerem Özdemir.