Swiss French are likely very similar to Haut-Savoyards (and people from Ain and also people that are from NW Savoie (north of "Bourget" lake)
I have to say that I had seen a Swiss French plotting very close to Austrians so don't know really. We lack samples, simply that.
According to Giemza :
https://media.springernature.com/ful..._Fig1_HTML.png
You do see the italian influence in Savoie (blue turquoise color)
Someone said that this map isn't reliable but this isn't true. There are 35 samples for each department and they used two panels separated on time, and tested on different SNPs (double randomized study), so it's very solid.