Simpson's completeness conjecture for the Lagrangians in the de Rham moduli space

Let $W^1(\bar\partial_0,\Phi_0)$ be one of the Lagrangians arising from the conformal-limit construction, and let $M_{dR}$ be the de Rham moduli space. Simpson's conjecture. The Lagrangians $W^1(\bar\partial_0,\Phi_0)$ are complete subvarieties of $M_{dR}$. The source states that the conformal-limit theorem enables the study of the geometry of these Lagrangians and was crucial in proving the conjecture; however, the supplied span itself gives no explicit resolution status, so the database records it as open pending verification.