Although I haven't done it mid June, I predict it will be an excellent time: very long daylight hours (15h+) and ‘ pleasantly busy' with mostly experienced hikers (the large bulk of first timers will appear from July onwards, when the summer holiday starts on most European schools).
I suspect most hikers will be French.
I would take a lightweight axe though. No doubt you’ll encounter snow on the higher stretches & passes (which I personally like as an ‘add on’ to the landscape).
The ‘old’ route I mentioned is absolutely not overgrown. Lots of hikers still use it. Just look for the over painted grey markings (and the big mountain it will lead you to, coming from the north). If using a gps, Wikiloc certainly holds some gps-tracks of the GR20 with the old route.
Using the other old route in the north, the one bypassing the refuge of d’Asco Stagnu, might be more tricky though, especially if snow still lingers around the mountain ridge you’ll follow. But the Dillon guidebook describes it sufficiently. And I recommend it strongly – excellent views and hardly any people. Bring sufficient water though. Apart from a small tarn, just before the final climb to the Cirque de la Solitude, there will be no water along this old route.
If you have a hotel reservation in the small ‘midpoint’ village of Vizzavone, you could sent a parcel towards them (post it from within France – of better – Corsica). Just inform the hotel beforehand as the usage of bounce boxes and the likes is virtually nonexistent in Europe.
Last time, I did the GR20 in 8.5 days, which is very feasible if going lightweight (it’s approx. 180km). I brought & carried about 90% all my food. Most refuges have a small shop where you can supplement you daily rations (e.g. pasta, tomato sauce, (old) bread, not to forget wine & beer).
I took a small single wall tent with me. Sleeping in refuges can be quit uncomfortable (hot during the night, bugs, noisy and often cramped as sleeping accommodation often are of the ‘Matratzenlager’ type). I would normally pitch my tent but still sleep outside ‘cowboy style’, unless there are mosquitos around. If it would start raining during the night, I can quickly retire to my tent.
Officially, wild camping is prohibited along the GR20. IMHO with good reason: due to its popularity, it would render the GR20 the Garbage Route 20. If you do camp wild, do it with much discretion: pitching the tent just before sunset and breaking up camp before sunrise. Mind you, possibilities for wild camping are often limited due to the ruggedness of the surroundings (ironically, you’ll often notice find the best wild camping spots where signs have been erected reminding hikers that wild camping is prohibited.
PM me if you need last minute information. Anyway, you're in for a fabulous trip.