unstability of a scroll 2

Question

For the purposes of this question I will work over $\mathbb{C}$. Consider on $T=\mathbb{P}^1$ the bundle $E=O_{T}^{\oplus 3}\oplus O_T(-1)^{\oplus 2}$ and $\mathbb{P}E_T$ the associated projective bundle on $T$. Let $[s,t]$ be coordinates on $T$. Let $w_0,w_1,w_2$ be a frame for $O_T^{\oplus 3}$ and $u_4,u_5$ sections of $O_T(-1)^2$ with poles at $s=0$. Set $w_4=su_4$, $w_5=su_5$, and let $X\subset \mathbb{P}E$ be the scroll given by the vanishing of the rank 2 minors of $$ \left( \begin{array}{} w_1 & w_2 & w_4\\ w_0 & w_1 & w_3 \end{array} \right) $$ Given $t\in T$, does there exist $m$ such that the $m$-Hilbert point of $X_t$ is not stable for the action of $SL$ on $H^0(X_t,O_{\mathbb{P}E}(1)|_{X_t})$?