The local orthogonality-preserver conjecture for full Hilbert C*-modules

Let $A$ be a $C^*$-algebra, and let $M(A)$ denote its multiplier algebra. Let $E$ and $F$ be Hilbert $A$-modules, with $E$ full, meaning that the linear span of

$$\{\langle x,y\rangle:x,y\in E\}$$

is dense in $A$. A map $\theta:E\to F$ is local if

$$\theta(x)a=0\quad\text{whenever}\quad xa=0\qquad (x\in E,\,a\in A).$$

Local orthogonality-preserver conjecture. If $\theta:E\to F$ is a $\mathbb C$-linear local map preserving orthogonality, namely

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

for all $x,y\in E$, then there is a central positive element $u\in M(A)$ such that

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

The conjecture extends known results for orthogonality-preserving module maps over standard $C^*$-algebras and for local maps over commutative $C^*$-algebras. The fullness assumption is necessary, while the general case for local orthogonality-preserving maps over $C^*$-algebras of real rank zero remains open in the source.