Fubini properties for filter-related σ-ideals

Let MF be the σ-ideal of meager sets in the Ellentuck topology related to a filter F on the set of natural numbers. We prove that if G is a filter with the Baire property and F is an analytic, non-countably generated p-filter, then the pair 〈MG,MF〉 does not have Fubini Property. In fact, there is a Borel set on the plane such that its vertical sections are in MF and the complements of its horizontal sections are in MG. This partially answers a question of Recław [Topology Appl., submitted for publication].