Let $Y$ be degree 5 index two prime Fano threefold. Let $\mathcal{E}$ and $\mathcal{Q}$ be the tautological sub and quotient bundle on $Y$. It is not hard to show that there is a short exact sequence: $$0\rightarrow\mathcal{E}\xrightarrow{p}\mathcal{Q}^{\vee}\rightarrow I_L\rightarrow 0$$, where $L$ is a line on $Y$. This is because one can show the map $p$ is injective by slope stability of $\mathcal{E}$ and $\mathcal{Q}^{\vee}$ and slope of them are $-\frac{1}{2},-\frac{1}{3}.$

But I was wondering whether the following exact sequence also exist or not? $$0\rightarrow I_L\rightarrow\mathcal{Q}\xrightarrow{\pi}\mathcal{E}^{\vee}\rightarrow 0$$ The problem to show the map $\pi$ is surjective is that the cokernel of $\pi$ could be rank zero sheaf, for example $\mathcal{O}_L$, then there is no contradiction by arguing with stability and slope.

The similar thing happened in Gushel-Mukai threefold, there is a short exact sequence $$0\rightarrow\mathcal{E}\xrightarrow{p}\mathcal{Q}^{\vee}\rightarrow I_C\rightarrow 0$$ with $C$ being a conic by the same stability argument and note that the zero locus of section of $\mathcal{Q}$ is either two points or a conic(in both special and ordinary GM case, with the argument slightly different)

But I was wondering whether the following exact sequence exist? $$0\rightarrow I_C\rightarrow\mathcal{Q}\xrightarrow{\pi}\mathcal{E}^{\vee}\rightarrow 0.$$ I thought it exists, but then I located a mistake in the argument, say the cokernel of $\pi$ could be $\mathcal{O}_L$ and the image of $pi$ can be a torsion free semistable sheaf $E\in M(2,1,5)$.

Maybe there is a very simple reason that such short exact sequence does not exist, which I miss it?