Let $f:X\rightarrow Y$ be a flat surjective morphism between reduced complex analytic spaces. Assume that $Y$ is locally irreducible (i.e. unibranch). We assume that $X$ (resp. $Y$) is pure-dimensional of dimension $m$ (resp. $n$). Let $\alpha$ be a smooth $(m-n+r,m-n+r)$-form ($r\geq 0$) with compact support on $X$. Is it true that the push-forward current $f_*[\alpha]$ is always represented by a form with continuous coefficients?

As far as I know, the following cases are known:

- When $X$, $Y$ are both smooth. 2. When $r=0$ by King's fibering theorem.

If the general case fails, what if we assume that $f$ is smooth?

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