I bought an LSU Purple from Whole Foods last year as well. Uppoted before the fall of 2014 into 8 gallon pot, and I have more figs on this tree then I ever expected. The only problem is the fruit is continually dropping. Some start to turn purple, but drop before they get that black-purple coloring. The tree is on my deck, and the deck is pure hot sun from 11a - 7p. Temp readings on the boards can reach over 130dgf. It needs to be watered every day or it droops. Just can't keep it from dropping figs. Still has about 4-6 dozen smallish figs on it, so I am hopeful. Had one or two that got near full ripe and the flavor was not much to brag about, figgy and a touch of sweet. The tree has beautiful shape, just not happy with how it drops fruit. Other trees on my deck are not dropping fruit (including my JH Adriatic, White Votato, and UNK OBX (found in the outer banks), White Triana). Would have thought a tree bred for the hot humid conditions of Louisiana would not struggle with the triple H's of the Delmarva Peninsula.
More to your question, when mine have reached the size you have shown in your picture, they are already starting to turn purple. But only a few have reached that size. Smallish figs on my LSU Purple. Good Luck!