The weak Albanese conjecture for compact Kähler C-pairs

Let $(X,D)$ be a C-pair, meaning a pair consisting of a complex space $X$ and a boundary divisor $D$ with Campana's orbifold multiplicities, and suppose that $X$ is compact Kähler. A weak Albanese is a normal analytic variety $Z^{\circ}$ equipped with a morphism

$$\operatorname{walb}(X,D)^{\circ}:X^{\circ}\to Z^{\circ}$$

that satisfies the universal property given in the source. Weak Albanese conjecture. A weak Albanese of $(X,D)$ exists; the variety $Z^{\circ}$ and the morphism $\operatorname{walb}(X,D)^{\circ}$ are unique up to unique isomorphism, and the group $\operatorname{Aut}_{\mathcal O}(X)$ acts on $Z^{\circ}$ so that $\operatorname{walb}(X,D)^{\circ}$ is equivariant. This is proposed for arbitrary compact Kähler C-pairs, including cases where an Albanese in the stronger sense cannot exist. Its status is not determined by the supplied source.