Geometric representation conjecture for the von Neumann continuous geometry

Let $L$ be the von Neumann example of a continuous geometry. A geometric representation of $L$ is a representation of the corresponding modular ortholattice by subspaces of a suitably orthogonal space. An elementary extension of the reals is a real closed field extending the real field in the relevant first-order language.

Geometric representation conjecture. The von Neumann example of a continuous geometry admits a geometric representation over an elementary extension of the reals.

The source notes that this example is proatomic and asks how far the assertion extends to abstract continuous geometries. The status of the displayed assertion is not resolved in the source.