Vx is the speed of maximum excess thrust = thrust available - thrust required (drag). If thrust available were not a function of speed, that would be the same as minimum drag speed (best glide for a glider). But since thrust available decreases with speed, Vx occurs at a lower speed than minimum drag speed. As the thrust available reduces with altitude, Vx increases to much closer to the minimum drag speed at the absolute ceiling of the aircraft.
Jets are closer to constant thrust systems -- power increases significantly with speed, and so the effect decreasing Vy is more significant than the increase in Vx. Both of those are simplifications: in fact thrust decreases and power increases with speed for both props and jets.
There's one more complication which is that the power required is a function of both IAS/CAS because it depends on drag, and TAS because power is drag times (true) speed. As altitude increases and TAS increases for a given IAS, the minimum power required will reduce. This effect will tend to reduce Vy with altitude, even for an engine that doesn't care so much about the thinner air.
Disagreed.
Imagine that you had an engine which offered constant thrust completely independent of speed or altitude. Such engines exist. Namely rockets.
If you start flying a rocket powered plane at, say, IAS and TAS of 50 knots, you would have some speed for stall, another speed for zero rate of climb, then Vx speed, then Vy speed (and associated maximum climb rate), then maximum speed for sustained level flight. Now go on and climb into stratosphere. When your speed is 250 knots TAS and still 50 kn IAS, your Vs, Vzrc, Vx, L/Dmax, Vy and Vmax would, in terms of IAS, be unchanged. It terms of TAS, they will have increased fivefold And your maximum climb rate also has increased 5 times.
The same applies when your speed is 500 knots TAS, except that Mach effects will affect the actual shape of polar curve (and your best L/D worsens with sound barrier wave drag). As well as at 5000 kn TAS (except that the hypersonic effects lead to further deterioration of L/D). As you approach 15 000 kn ground speed, centrifugal forces increase and your weight vanishes.
So - for an engine whose thrust is constant, there is no such a thing as ceiling.
Now, if you have an engine whose thrust DOES derate with altitude and airspeed, because it does take in air, as the plane climbs towards it ceiling, in terms of IAS we can expect that Vs would be unchanged, Vzrc and Vx would increase, while Vy and Vmax would decrease. Above the ceiling, the whole polar curve would be below zero, Vs would be unchanged, there would be no Vzrc or Vmax (no crossing of polar curve and zero rate of climb), and there would be Vy and Vx - with Vy smaller than Vx.
And exactly at the ceiling, the polar curve would touch the zero rate of climb at one speed, which is equally Vzrc, Vx, Vy and Vmax.
I should expect Vy to decrease with altitude in terms of IAS. But what should happen to Vy in TAS terms? I see no reason why a plane whose thrust does decrease with speed and altitude, but does so only slowly (low-bypass turbofan, afterburner...) could not have TAS Vy increasing all the way to its ceiling where it is overtaken by Vx?