Seshadri constants on some flag bundles

Let $X$ be a smooth complex projective curve and let $E$ be a vector bundle on $X$ which is not semistable. We consider a flag bundle $\pi: \text{Fl}(E) \to X$ parametrizing certain flags of fibers of $E$. The dimensions of the successive quotients of the flags are determined by the ranks of vector bundles appearing in the Harder-Narasimhan filtration of $E$. We compute the Seshadri constants of nef line bundles on $\text{Fl}(E)$.