Mount Charleston Peak winter season? (Or trail running near Vegas)

Anybody familiar with the winter / snow season for Mt Charleston? I may be in the area mid-/late-March and was looking at running to the peak one of the days. Odds on the trails being snow/ice free enough and open for running? Looks like the Cathedral Rock picnic area is closed in the winter but I didn't find expected dates for that to run through. Looks like snow free is generally June-October so I'm guessing late March is unlikely to work out.

If it does, anybody have a recommendation on doing the loop clockwise or counter clockwise? Either direction better or more runnable?

Edited title - new post below for general trail running ideas.

John Wilder wrote: The trails themselves don’t close, but they will definitely still have snow at the higher elevations that time of year.

If the snow level is below the ridge line, I’d recommend the South Loop out and back, as the North Loop has a couple spots on the ridge that if you slipped, you might not be able to arrest your fall...

Edit- not sure how runnable they will be, but I’m spending the winter hiking and snowshoeing up there this year, so dm me when it gets closer and I’ll give you an update.

Thanks - I'll check in closer to it! Good tip on not sliding to my death. I do have spikes but am not looking for an ice fest :) I'd look for other options at that point.

Bumping back up. Doubtful Charleston will be runnable, so does anyone have other trail running suggestions? I'd love to get in longer climbs than what is near me in PA (so over 1000') but favorite scenic routes out there are good too. I'll be on the strip so extra bonus if a bus or something can get me out to the start. I'll be there the last week in March with no vehicle.

Google Frenchman mountain, probably have to rent a car.
It’s Northeast just out of town. Definitely desert.