Before revision:  Furthermore, we obtain a PTAS in case all the input L-frames are anchored at the diagonalfrom one side. Next, we consider the version, where all input L-frames intersect a vertical line , and prove APX-hardness of this version.
After revision:  Furthermore, we obtain a PTAS in case all the input L-frames intersect the diagonal, or the horizontal segments of the L-frames intersect a vertical line , and prove APX-hardness of this version.