Running at very high altitudes (4400m/14400ft.)

Hi

I'm spending a week in western Sichuan, in the high Tibetan plateau. We'll be acclimating at around 3000m for one night before reaching the city of Litang at 4400m. I'm intrigued about attempting a few runs at this altitudes. Beyond hydration and sunscreen are there safety concerns as long as I "feel good" and don't suffer any altitude sickness before the run?

Could running set altitude sickness off and leave me much worse off for the remainder of the visit?

I have previously hiked at similar altitudes without any problems.

Thanks!