Thank you for spending some time to keep the community posted @francois-techene 
I do have a question regarding the proposed UX: are all these gestures expected to be performed with a thumb? It seems some parts of a 5 or 5.5" screen are hard to reach when using in single handed mode.
Plus, this is not exactly design, but if the phone is designed for a one handed use, the case should probably be made of “grippy” material. The iPhones are a perfect counter example of what should be done for the case, they tend to slip a lot. I actually dropped mine a few times.