A Dolbeault lemma for temperate currents

We consider a bounded open Stein subset $\mathrm{\Omega }$ of a complex Stein manifold $X$ of dimension $n$. We prove that if $f$ is a current on $X$ of bidegree $(p,q+1)$, $\bar{\mathrm{\partial }}$-closed on $\mathrm{\Omega }$, we can find a current $u$ on $X$ of bidegree $(p,q)$ which is a solution of the equation $\bar{\mathrm{\partial }}u=f$ in $\mathrm{\Omega }$. In other words, we prove that the Dolbeault complex of temperate currents on $\mathrm{\Omega }$ (i.e. currents on $\mathrm{\Omega }$ which extend to currents on $X$) is concentrated in degree $0$. Moreover if $f$ is a current on $X=C^n$ of order $k$, then we can find a solution $u$ which is a current on $C^n$ of order $k+2n+1$.