Let $k$ be an algebraically closed field of positive characteristic and $f:X \to S$ be a smooth, flat, projective morphism between noetherian $k$-schemes. Assume that $S$ is a non-singular quasi-projective variety. Let $\mathcal{F}$ be a locally free sheaf on $X$. Fix a polarization $L$ on $X$ and a Hilbert polynomial $P$. Denote by $\mathrm{Quot}_{X/\mathcal{F}/S}^P$ the relative Quot-scheme parametrizing coherent quotients of $\mathcal{F}$ with Hilbert polynomial $P$.

There exists a natural morphism $\pi:\mathrm{Quot}_{X/\mathcal{F}/S}^P \to S$ and for each point $x \in \mathrm{Quot}_{X/\mathcal{F}/S}^P$, there exists an induced map between tangent spaces $T_x\pi:T_x\mathrm{Quot}_{X/\mathcal{F}/S}^P \to T_{\pi(x)}S$. Is it true that for a general (closed) $x \in \mathrm{Quot}_{X/\mathcal{F}/S}^P$, we have $\dim \mathrm{Im}(\pi) \le \mathrm{rk}(T_x\pi)$? Any hint/reference on this topic will be very welcome.

N.B. If it helps, one can assume that $X$ is of the form $C \times S$, for a non-singular curve $C$ over $k$ of genus at least $2$ and $f$ is projection onto the second coordinate.