Naive dimension conjecture for secant loci of scrolls over curves

Let $C$ be a nonhyperelliptic curve of genus $g\geq 9$. Let $E\to C$ be a vector bundle of rank $r$ and slope at most $g-2$ such that the tautological line bundle $\mathcal O_{\mathbb P E}(1)$ is very ample. For $e\leq h^0(C,E)-2r$, consider the secant locus $Q^{e-f}_e$ and the corresponding locus $(H^{e-f}_e)_{\mathrm{nd}}$ of $\pi$-nondefective subschemes. Naive dimension conjecture. One has

$$\dim Q^{e-f}_e\leq \dim\left(H^{e-f}_e\right)_{\mathrm{nd}}\leq e-f-r.$$

This conjecture gives the expected dimension bound suggested by examples involving direct sums of line bundles; it concerns the nondefective part of the secant locus and is posed for nonhyperelliptic curves of sufficiently large genus under the stated positivity and numerical hypotheses.