I also have a supposed Improved Celeste, leaf shape and fruit match what you have pictured. Like you said it is a very good tasting fig. Mine is only 2nd season and fruit drop was about 60-70%. It has taken a break now but it still has many green unripe figs on it and it is not dropping as many. It seems like they dropped when they showed a little color, the ones that dropped turned more grayish then brown and fell. I will give one more season. If fruit drop decreases then it will be a keeper.