This is a cross-post of two recent questions at math.stackexchange without answers: Q1 and Q2.
A boolean function on an $n$-dimensional hypercube is linearly separable when the convex hulls of the points evaluating to $0$ and $1$ respectively are disjoint.
A necessary condition is that any rectangle (not only the faces) formed by $4$ points of the hypercube is linearly separable.
Is this condition also sufficient?
I have tested that the number of linearly separable boolean functions (OEIS A000609) on $n \le 4$ variables is equal to the number of hypercubes with all rectangles formed by $4$ vertices linearly separable (these ones counted using this answer and this software). Therefore the above condition is proven sufficient for $n \le 4$.
Using the suggestion on the below comment by @Fedor Petrov I was able to prove the conjecture for $n \le 5$ using this updated software which took about $8$ minutes on my computer, which decreases to about $5$ minutes if I test the next boolean function on the rectangle where the previous one failed the linear separability test.
UPDATE 2024-02-13
Using this software I was able to compute the expected value of $15028134$ for $n=6$ in $22$ hours, so now the conjecture is proven for $n \le 6$. Later this other software did it even better in $2$ hours and $45$ minutes. However now it looks impossible to go further searching for a possible counterexample, unless a more advanced algorithm, exploiting some structure of the problem and with much lower computational complexity, is found.
UPDATE 2024-02-16
A new software, exploiting some symmetry, computed the case $n = 6$ in only $2$ seconds and the expected value $8378070864$ for $n = 7$ in $1$ hour and $19$ minutes. Therefore now the conjecture is true for $n \le 7$.
