Linear orthogonality preservers of Hilbert $C^*$-modules over $C^*$-algebras with real rank zero

Let $A$ be a $C^*$-algebra with real rank zero. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $\theta : E\to F$ is a linear map preserving orthogonality, i.e.,

$$\langle \theta (x),\theta (y)\rangle \ =\ 0$$ $$\quad \text {whenever}\quad \langle x,y\rangle \ =\ 0.$$

We show in this article that if $\theta$ is an $A$-module map (not assumed to be bounded), then there exists a central positive multiplier $u\in M(A)$ such that

$$\langle \theta (x), \theta (y)\rangle \ =\ u \langle x, y\rangle \qquad (x,y\in E).$$

In the case when $A$ is a standard $C^*$-algebra, when $A$ is a real rank zero properly infinite unital $C^*$-algebra, or when $A$ is a $W^*$-algebra, we also get the same conclusion with the assumption of $\theta$ being an $A$-module map weakened to being a local map.