Albanese morphism of log smooth klt compact Kähler manifold with nef log anticanonical divisor

Let $(X, \omega)$ be an n-dimensional compact Kähler manifold. Let $D=\sum (1-\beta_j) Y_j=\sum (1-\beta_j) [s_j=0]$ a divisor with simple normal crossings with $\beta_j \in ]0,1[$ such that $-(K_X+D)$ is nef. We show that its Albanese map is submersion outside an analytic set of codimension larger than two with connected fibres.