Let $X$ be a smooth projective variety with polarization $\mathcal{O}_X(1)$. For fixed Chern class $r,c_1,c_2....$, do we have an OPEN embedding $$M^{ss}(X,r,c_1...)_{vec}\subset M^{ss}(X,r,c_1...)$$ of moduli space (Gieseker)-semistable vector bundles into the moduli space of semistable sheaves with the above Chern classes? ( I guess I should assume there is such a vector bundle to start with)
Also, since $\mu$-stable implies stable, is the moduli space of $\mu$-stable vector bundles with the above Chern character openly embedded in $M^{ss}(X,r,c_1...)_{vec}$?
Thanks!