That looks like relatively smooth tile. Okay. I'd use a 4" wallpaper razor scraper with a new blade. Run the scraper to the edge of the little "island" of grout and swing the handle left and right while pressing forward.
PLEASE EXCERCISE EXTREME CAUTION AND NEVER PUT ANY BODY PARTS IN FRONT OF THE BLADE!
Once you get most of the grout removed, you can finish the tiny remnants with the haze remover that was working for you. Now, beware that the scraper is likely to leave small metal deposits on the tile, most commonly referred to as "metal marks". These can be very easily removed with POWDERED "Bar Keepers Friend" cleaner. Use a damp paper towel with a tiny bit of the powder and it'll come off within several seconds.