Why was the early proposal involving interstellar dust failing to solve Olbers' Paradox?

The intervening dust would eventually absorb so much energy that it would heat up and begin radiating energy itself.

One early attempt to resolve the paradox suggested that intervening interstellar dust absorbs the light emitted by distant stars, thus preventing the sky from blazing. However, this solution is insufficient because, over vast stretches of time, this dust would continuously absorb energy from the stars. This accumulation of absorbed energy would cause the dust layer itself to heat up significantly until it begins glowing across the entire electromagnetic spectrum, effectively replacing the starlight with a dim glow from the dust, meaning the sky would not be truly dark.