Linear transformations of monotone functions on the discrete cube

For a function f : { 0 , 1 } n → R and an invertible linear transformation L ∈ G L n ( 2 ) , we consider the function L f : { 0 , 1 } n → R defined by L f ( x ) = f ( L x ) . We raise two conjectures: First, we conjecture that if f is Boolean and monotone then I ( L f ) ≥ 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.