Complete Bullshit. if that were in any way true climbing Everest would be impossible.
At 6,000 ft (which is where the town of Jackson is), i doubt that the OP will notice much of a difference, but at 10,000 ft (the approx altitude of the Headwall hike), he will definitely notice a difference.
Regarding adjusting to altitude:
Altitude sickness - Wikipedia, the free encyclopedia
The general rule of thumb is to not ascend more than 300 metres (1,000 ft) per day to sleep. That is, one can climb from 3,000 (10,000 feet = 70 kPa) to 4,500 metres (15,000 feet = 58 kPa) in one day, but one should then descend back to 3,300 metres (11,000 feet = 67.5 kPa) to sleep. This process cannot safely be rushed, and this is why climbers need to spend days (or even weeks at times) acclimatising before attempting to climb a high peak.
So, even someone living in Denver would require about 5 days for full acclimatization to physical activity at 10,000 ft. One's mountain sickness may go away after a day or two, but you will still be huffing and puffing on that hike up the Headwall until you've been around for a week unless you already have godly cardio.
Also, people climbing Everest spend weeks doing it. Not days.
Anecdote: I happened to hike up a catwalk at Snowbird with a girl I was giving an impromptu lesson to. We had both just flown in the night before from Atlanta and were staying in the same house. 100 yards into our hike I was wheezing and she wasn't. This despite the 5-7 miles a week I had been running in preparation for the trip.
When I asked her why she wasn't gasping for breath like I was, she informed me that she had been training for a marathon for the past few weeks.
I could tell you other stories about other Atlantans in Breckenridge (elevation = ~9600 ft), but the short version is that you still won't be at 100% in just a few days at altitude.