An order-eight counterexample to the orthogonal extremizer conjecture

Let $L_8$ be the $8\times8$ unit lower triangular matrix with every entry below the diagonal equal to $-1$, and let $Q_8$ be its orthogonal factor in the QR factorization with positive diagonal in the triangular factor. Changing the $(8,2)$ entry of $L_8$ from $-1$ to $0$ and taking the orthogonal factor of the same QR convention gives an explicit $\widetilde Q\in O(8)$ for which partial pivoting, choosing the first available row in every tie, gives $\rho_{\mathrm{PP}}(\widetilde Q)=5272/63>\sqrt{17948132/2601}=\rho_{\mathrm{PP}}(Q_8)$. Hence the supremum of the partial-pivoting growth factor over orthogonal matrices and admissible pivot paths exceeds $\rho_{\mathrm{PP}}(Q_8)$ already at order eight. The proof is in exact arithmetic: it gives integer-column descriptions of both orthogonal matrices, their no-exchange LU factorizations and the rational squared maxima of every active Schur complement. It does not determine the true supremum.