Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its pullback to the normalization of $X$. Suppose $\pi^{*}(F)$ is Gieseker semistable, then is $F$ Gieseker semistable?

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its pullback to the normalization of $X$. Suppose $\pi^{*}(F)$ is Gieseker semistable, then is $F$ Gieseker semistable?

Is it true when $X$ is irreducible?