Linear transformations of monotone functions on the discrete cube

For a function f : {0, 1} → R and an invertible linear transformation L ∈ GLn(2), we consider the function Lf : {0, 1} → R defined by Lf(x) = f(Lx). We raise two conjectures: First, we conjecture that if f is Boolean and monotone then I(Lf) ≥ I(f), where I(f) is the total influence of f . Second, we conjecture that if both f and L(f) are monotone, then f = L(f) (up to a permutation of the coordinates). We prove the second conjecture in the case where L is upper triangular.