Kobayashi's standard-form conjecture for reductive homogeneous spaces

Let $G$ be a Lie group and let $H$ be a reductive subgroup. A compact form of $H\backslash G$ is a compact quotient by a properly discontinuous group action. Suppose that $L$ is a Lie subgroup of $G$ such that

$$HL=G,$$

$H\cap L$ is compact, and $\Gamma$ is a uniform lattice in $L$. Such a quotient

$$H\backslash G/\Gamma\cong (H\cap L)\backslash L/\Gamma$$

is called a standard form; it is enough more generally that $HL$ be cocompact in $G$. Kobayashi's standard-form conjecture. If $H\backslash G$ possesses a compact form, then it possesses a standard form.