Riemann-Hilbert morphism RH(X) for a smooth complex algebraic variety

For a smooth complex algebraic variety X, one can define a (Betti) moduli space of complex local systems on X, M_B(X). Choosing a nice compactification with boundary a simple normal crossings divisor, one also defines a (de Rham) moduli space of logarithmic flat connections, M_DR(X). These are algebraic varieties related by an analytic morphism of their underlying complex manifolds, RH(X): M_DR(X) --> M_B(X). In the projective case, this is a biholomorphism. In particular cases, it also boils down to the exponential map from above. My conjecture is: the Riemann-Hilbert morphism RH(X) for a smooth complex algebraic variety should satisfy the Ax-Lindemann-Weierstrass property.