Google PageSpeed & ImageMagick JPG compression

Given a user uploaded image, I need to create various thumbnails of it for display on a website. I'm using ImageMagick and trying to make Google PageSpeed happy. Unfortunately, no matter what quality value I specify in the convert command, PageSpeed is still able to suggest compressing the image even further.

Note that http://www.imagemagick.org/script/command-line-options.php?ImageMagick=2khj9jcl1gd12mmiu4lbo9p365#quality mentions:

For the JPEG ... image formats, quality is 1 [provides the] lowest image quality and highest compression ....

I actually even tested compressing the image using 1 (it produced an unusable image, though) and PageSpeed still suggests that I can still optimize such image by "losslessly compressing" the image. I don't know how to compress an image any more using ImageMagick. Any suggestions?

Here's a quick way to test what I am talking about:

assert_options(ASSERT_BAIL, TRUE);

// TODO: specify valid image here
$input_filename = 'Dock.jpg';

assert(file_exists($input_filename));

$qualities = array('100', '75', '50', '25', '1');
$geometries = array('100x100', '250x250', '400x400');

foreach($qualities as $quality)
{
    echo("<h1>$quality</h1>");
    foreach ($geometries as $geometry)
    {
        $output_filename = "$geometry-$quality.jpg";

        $command = "convert -units PixelsPerInch -density 72x72 -quality $quality -resize $geometry $input_filename $output_filename";
        $output  = array();
        $return  = 0;
        exec($command, $output, $return);

        echo('<img src="' . $output_filename . '" />');

        assert(file_exists($output_filename));
        assert($output === array());
        assert($return === 0);
    }

    echo ('<br/>');
}