Funcoids are filters? Conjecture II

Earlier I have conjectured that the set of funcoids is order-isomorphic to the set of filters on the set of finite joins of funcoidal products of two principal filters. For an equivalent open problem I found a counterexample.

Now I propose another similar but weaker open problem:

Conjecture Let U be a set. The set of funcoids on U is order-isomorphic to the set of filters on the set \Gamma (moreover the isomorphism is (possibly infinite) meet of the filter), where \Gamma is the set of unions \bigcup_{X\in S}(X\times Y_X) where S is a finite partition of U and Y\in \mathscr{P} U for every X\in S

The last conjecture is equivalent to this question formulated in elementary terms. If you solve this (elementary) problem, it could be a major advance in mathematics.

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