Within their JS I found a detectPrivateMode function which seems roughly based around this gist: https://gist.github.com/cou929/7973956

The key lines that trigger detection are these:
    var is_private;

    if (window.webkitRequestFileSystem) {
        window.webkitRequestFileSystem(
            window.TEMPORARY, 1,
            function() {
                is_private = false;
            },
            function(e) {
                console.warn(e);
                is_private = true;
            }
        );
    }