Existence of non-normal EP operators with prescribed range

Let $\mathcal H$ be a Hilbert space and let $\mathcal W\subseteq\mathcal H$ be a closed subspace. An operator $T$ on $\mathcal H$ is EP when its range agrees with the range of its Moore–Penrose inverse, and $\mathcal R(T)$ denotes its range.

Existence conjecture. There exists a non-normal EP operator $T$ on $\mathcal H$ such that

$$\mathcal R(T)=\mathcal W.$$

The statement asserts that every closed subspace of a Hilbert space can occur as the range of a non-normal EP operator. The supplied text does not indicate whether this assertion is known or unresolved.