Failure of the semi log canonical Abundance for compact Kähler threefolds

In this article we show that the semi log canonical abundance for compact Kähler varieties fails in dimension $3$. More specifically we construct a counterexample of a compact Kähler (irreducible) slc threefold $(X,0)$ such that $K_X$ is nef and $\kappa (\tilde{X},K_{\tilde{X}}+\tilde{D})=0$, where $\mu :(\tilde{X},\tilde{D})\to X$ is the normalization morphism, but $K_X$ is not semiample. On the other hand, we show that if we start with a compact Kähler semi-dlt pair, then the abundance does hold, i.e., if $(X,\mathrm{\Delta })$ is a compact Kähler sdlt pair of dimension $3$ such that $K_X+\mathrm{\Delta }$ is nef, then it is semiample. We also show that if $(X,\mathrm{\Delta })$ is a compact Kähler slc pair of dimension $3$, $K_X+\mathrm{\Delta }$ is nef, and $\kappa (X_i^{\mathrm{\prime }},{\mathrm{\Delta }}_i^{\mathrm{\prime }}+D_i^{\mathrm{\prime }})>0$ for all $i$, where $\mu :\bigsqcup (X_i^{\mathrm{\prime }},{\mathrm{\Delta }}_i^{\mathrm{\prime }}+D_i^{\mathrm{\prime }})\to (X,\mathrm{\Delta })$ is the normalization, then $K_X+\mathrm{\Delta }$ is semiample.