Conjecture: Distributivity of a lattice of funcoids is not provable without axiom of choice

Conjecture Distributivity of the lattice $latex \mathsf{FCD}(A;B)$ of funcoids (for arbitrary sets $latex A$ and $latex B$) is not provable in ZF (without axiom of choice).

It is a remarkable conjecture, because it establishes connection between logic and a purely algebraic equation.

I have come to this conjecture in the following way:

My proof that the lattice of funcoids is distributive uses the fact that it is an atomistic lattice. That $latex \mathsf{FCD}(A;B)$ is an atomistic lattice in turn uses the fact that the lattice of filters on a set is atomically separable and it follows from the fact that the lattice of filters on a set is an atomistic lattice.

But that the lattice of filters on a set is an atomistic lattice cannot be proved without axiom of choice. So the axiom of choice is used in my proof of distributivity of the lattice of funcoids.