Is a subspace containing a splitting subspace a splitting subspace?

If V, A and B are three closed subspaces of we say that V is a splitting subspace for A,B if and only if A and B are conditionally orthogonal given V. If V is a splitting subspace for A,B, we shall say that V splits A,B. Rozanov [Rozanov, Y.A., 1979. Stochastic Markovian Fields. In: Developments in Statistics, vol. 2. Academic Press, New York, p. 205] observes that A[perpendicular]BV does not imply that the closed subspace W[superset of or equal to]V splits A,B. However, no example is provided. In this note we provide one.