Property tau Need Not Survive Intersections of Normal Subgroups

We construct a family of finite-index normal subgroups of a fixed finitely generated free group which has property (tau), whereas its prefix-intersection chain does not. The construction uses Kassabov's bounded-degree alternating-group expanders. Two quotient maps have identical expanding generators and send one additional generator to the identity and a 3-cycle. Their joint image is the full product Alt(N) x Alt(N), but its action on N squared ordered pairs has a centered diagonal test function with lazy Rayleigh quotient 3/[2r(N-1)]. Equal-fiber pullback transfers the obstruction to the regular quotients and to the prefix chain. This gives a complete negative answer to Question 9 in Miklos Abert's 2010 list (AMR-011-0009 in UnsolvedMath v1.6.0). The source archive includes the English LaTeX manuscript, full argument, self-audit, two exact finite checkers and their outputs. Kassabov's expansion theorem is an established input, not a new result of this paper. The construction is distinct from the nonnormal-subgroup counterexample and does not contradict the fixed-subgroup intersection theorem of Abert and Elek. No claim is made about Mimura's more restrictive LEF-approximation question. This AI-assisted, self-audited preprint is unrefereed. A bounded primary-literature search did not locate this exact construction, but novelty and absolute priority are not certified. No independent specialist review or proof-assistant verification is claimed.