I know some times they don’t like the use of a queen excluder but when they need the space they will have no problem with it. Depending on where you live you may be entering a dearth. (Mid July into September) Honeybees will be very reluctant to draw out comb without a strong nectar flow. I would inspect your second Deep box again to see if they finished drawing out the rest of that comb. You may have to feed sugar syrup to stimulate comb construction if you don’t plan on using the honey this year, or you can wait until the fall flow starts in late August to mid September depending on where you live. If it is a strong flow they will begin to use your super.
You also have to realize that from now until next spring your colony will be shrinking in size as the queen lays less eggs. So if they don’t need that extra space they won’t use it. Best of luck, Happy Beekeeping.