Resolving Das–Mohanty's concluding conjecture (ELA 40 (2024), p. 488) on the singular squared-distance Laplacian

Both clauses of Das–Mohanty's concluding conjecture in Electronic Journal of Linear Algebra 40 (2024), p. 488, for equation (3.7), hold: for a singular squared-distance matrix of a complete multipartite graph, the positive eigenvalues of its Laplacian-like matrix are one quarter and one, with multiplicities N−t and h−1, and its inertia is (N−s−1,1,s). A rank-one representation gives the inertia for every nonzero-cofactor case, corrects Theorem 4.4(ii) with counterexample K_{1,1,3,3}, and determines the remaining singular spectrum. The same congruence argument proves the singular-input inertia and continuous-family statements of Lucid's MathOverflow question 513954, revision 2.