Finite-test homotopy equivalence conjecture for finite-dimensional CW complexes

Let $K$ be a CW complex of finite dimension, and let $f:K\to K$ be a map such that, for every map $g:L\to K$ from a finite CW complex $L$, one has

$$f\circ g\sim g.$$

Finite-test homotopy equivalence conjecture. The map $f$ is a homotopy equivalence.

The paper notes that, if valid, this conjecture would be stronger than its main theorem. Its resolution is not given in the supplied text.