Adjoint positivity and embedding conjecture for weakly pseudoconvex complex spaces

Let $X$ be a non-compact weakly pseudoconvex complex space of pure dimension $n$, let $\pi:\widetilde{X}\longrightarrow X$ be a canonical desingularization, and let $L\longrightarrow X$ be a holomorphic line bundle. Assume that $L$ is positive. Write $\omega_X^{GR}=\pi_*\omega_{\widetilde{X}}$ for the Grauert–Riemenschneider canonical sheaf. Adjoint positivity and embedding conjecture.

(a) Does there exist an integer $m\geq n(n+1)/2$ such that

$$K_{\widetilde{X}}\otimes\pi^*L^{\otimes m}$$

is semi-ample?

(b) Does there exist an integer $m\geq n(n+1)/2$ such that

$$\omega_X^{GR}\otimes L^{\otimes m}$$

is ample? Furthermore, is

$$(\omega_X^{GR}\otimes L^{\otimes m})^{\otimes(n+2)}$$

very ample, so that the linear system

$$|(\omega_X^{GR}\otimes L^{\otimes m})^{\otimes(n+2)}|$$

gives a holomorphic embedding of $X$ into $\mathbb{P}^{2n+1}$? This extends the corresponding embedding theorem for smooth non-compact weakly pseudoconvex manifolds to singular complex spaces; the semi-ampleness, ampleness, and embedding assertions remain open in the stated generality.