We consider a variant of Vince's half-frequency conjecture for union-intersection systems in which an entire system must be nontrivial, while a system is declared trivial only when all its blocks are singletons or when its only block is the whole ground set; every such trivial system is stipulated to be irreducible. For nontrivial systems, irreducibility retains Vince's two-part partition criterion, and complete irreducibility is tested on the system and all subsystems induced by actual blocks. We give a system on 23 points, generated by 17 three-element sets, whose union-intersection closure consists of 111 distinct blocks. Every point belongs to at most 55 blocks, yet the system is nontrivial and completely irreducible under the stipulated convention. A normal-form count, a membership-preserving quotient, and explicit certificates for the induced subsystems make the argument self-contained. This is a counterexample to the explicitly modified conjecture, not an assertion that the original wording of Vince's Conjecture 5 has a unique settled interpretation.