Linearity conjecture for translation generalized quadrangles

Let $(\mathcal{S},A)$ be a translation generalized quadrangle, with kernel $k=\operatorname{end}^{\mathrm{t}}(A,A)$. A projective representation means an embedding of the quadrangle in projective space over a division ring.

Linearity conjecture. Every translation generalized quadrangle has a projective representation; equivalently, the kernel $k$ is a division ring.

This is the central obstacle in the theory of general translation generalized quadrangles. The conjecture was established for finite translation generalized quadrangles and, according to the source, is solved in positive characteristic; the characteristic-zero case remains open.