Two propositions and a conjecture

I added to Funcoids and Reloids article the following two new propositions and a conjecture:

Proposition (\mathsf{FCD}) (f\cap^{\mathsf{RLD}} ( \mathcal{A}\times^{\mathsf{RLD}} \mathcal{B})) = (\mathsf{FCD}) f \cap^{\mathsf{FCD}} (\mathcal{A}\times^{\mathsf{FCD}} \mathcal{B}) for every reloid f and filter objects \mathcal{A} and \mathcal{B}.

Proposition ( \mathsf{RLD})_{\mathrm{in}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{in}} f) \cap^{\mathsf{RLD}} ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}) for every funcoid f and filter objects \mathcal{A} and \mathcal{B}.

Question ( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f) \cap^{\mathsf{RLD}} ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}) for every funcoid f and filter objects \mathcal{A} and \mathcal{B}?

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