Parabolic compatibility conjecture for the integrable tame harmonic bundle

Let $X$ be a compact Riemann surface, let $P\subset X$ be a finite set, and let $(V,\nabla_V)$ be a semisimple holomorphic flat bundle on $X\setminus P$. Let $(H,D_V,h)$ be the tame harmonic flat bundle associated with it, and let $U_0$ be the endomorphism associated with the integrable flat harmonic bundle. Parabolic compatibility conjecture. If $(H,D_V,h)$ is integrable, then $U_0$ is compatible with the parabolic filtration defined by $h$ near each puncture. This is a punctured-curve refinement of the integrability theory for tame harmonic bundles; the supplied source excerpt gives no resolution of the assertion.