Reports on the official Switzerland snow website back in April were that it wasn’t a very big snowpack in that part of Switzerland compared with historical average. So could be more walking than usual, and perhaps the crevasses not as well-bridged as some years (? greater risk of punching thru ?)
Hate to bring up another detail in the context of this grand plan, but have you checked the weather forecast for next week?
If it were me, I’d be thinking more about hiking + climbing in the Calanques, bicycling down by the Mediterranean coast—maybe hoping for a 2-3 day favorable window to catch a section of the HR - (or a section of one of the 30 to 40 other worthwhile multi-day ski routes in the Alps that might get better weather?).
Since there’s not much skiing on the HR—mostly a “view” tour—I don’t see much point in doing it following a GPS in the fog—and not seeing most of the views.
If you can get some airline tickets to California, the Sierra mountains there have some nice multi-day ski traverses (and base camps and single-day tours + rock climbing / bouldering)—and the weather models for Sierras are showing stable weather for next week.