Can we redirect the thermal radiation of an object back to itself to heat it more?

Question

Imagine we have a tungsten wire that we heat up to 1000 kelvins inside a tube with a special coating that reflects 99.99% of tungsten's emitted light due to thermal radiation back to the wire.

My intuition says that the wire will absorb some of the redirected light and it will cause it to heat up and release more light creating a positive feedback loop till it melts and evaporates(if the special coating doesn't do it first).

Are there any fundamental violations in my reasoning?

Edit:

For clarification i assumed that an electric current was used to heat up the wire to 1000 kelvins and the same current is always applying through the whole process.

Answer 1

Are there any fundamental violations in my reasoning?

Yes, conservation of energy.

The energy gained from reabsorbing radiation can't be higher than the energy lost to radiation.

To address your edit: If you have some power (say 1000 W) coming in, you will reach thermal equilibrium when radiative heat losses equal 1000W.

By putting a reflective coating and recouping 99% of losses, then the wire will heat up to whatever temperature allows it to radiate at $\frac{1000}{0.99} = 100,000$W, so that the supplied 1000W keep it in equilibrium.

Of course the more efficient the reflective coating is, the more you have to worry about the coating itself being able to withstand the head and not radiate heat itself.