Linearity for translation generalized quadrangles

Let $\Gamma^x$ be an infinite translation generalized quadrangle with translation group $T$. Let $z$ be an affine point, meaning a point not collinear with $x$, and let $\{U\}_z$ be the set of lines incident with $z$. For each such line $V$, put $v:=\operatorname{proj}_V x$. Define $\mathbb{K}$ to be the set of endomorphisms of $T$ that map every $T_V$ into itself. Then $\mathbb{K}$ is a ring with multiplicative identity and without zero divisors, and $T$, $T_V$, and $T_v$ are left $\mathbb{K}$-modules. Linearity for TGQs. The ring $\mathbb{K}$ is a skew field. This would provide the expected linearity result for infinite translation generalized quadrangles, analogous to the finite case, where the corresponding embedding in projective space over a skew field is known.