Correlated fractal percolation and the palis conjecture

Let F1 and F2 be independent copies of one-dimensional correlated fractal percolation, with almost sure Hausdorff dimensions dimH(F1) and dimH(F2). Consider the following question: does dimH(F1) + dimH(F2) > 1 imply that their algebraic difference F1 ? F2 will contain an interval? The well known Palis conjecture states that ‘generically’ this...