The Illustrator distortion tool comes quite close to the desired result, especially when creating the envelope using an existing shape. However, the dots don't stay in horizontal lines and the distortion is curved to the right side instead to the left side.
I created a pattern swatch for the dots and adjusted the rectangle using it so that exactly the same number of rows and columns of these dots is used as in the template image.
Is it possible to fine tune the distortion that follows the given shape? Or should I give up the shape-approach and manually distort using a mesh?