The last time I flew to Wisconsin in the Citation V, center asked us if we could be 5,000 foot higher in about 3:30. Contract pilot flying said sure and pointed the nose up. We were mid 30s at the time. We lost airspeed as we climbed and eventually called Center to say it would be another minute when IAS dropped below 140 IAS. Afterward he pointed out that the AP in this plane would climb/descend using VS or IAS mode. Pointed out how VS (vertical speed) mode could lead to a stall and said some airlines prefer IAS (Indicated Air Speed). Easy to see how if the plane is in vertical speed mode, air speed could drop and lead to a stall on AP (which is supposed to be used in RVSM airspace). On the check ride, one maneuver is an AP stall and recovery. If the V gets past the stick shaker and fully stalls, it could be a handful. I would imagine swept wing planes would be worse.

Problem is in FLC/IAS climb mode in the 30's the AP trying to hold a Mach number will lead to little to no VS at times. To keep climbing steadily you have to switch to VS set at 400 to 500 fpm and monitor your airspeed and AOA. If you can't climb at a few hundred FPM and maintain a safe airspeed margin then you should level off until you get lighter.

You have to know your plane and the performance you can expect at a given altitude, ISA, and weight.