Non-torsion points for the Osada polynomial family

Let $n\geq 5$ be odd, let $0\neq d\in\mathbb Z$ be chosen so that $g(X)=X^n-X-1+d^2$ has distinct roots, and let $C/\mathbb Q$ be the smooth projective hyperelliptic curve defined by $Y^2=g(X)$, embedded into its Jacobian using the point at infinity. Let $L_f/\mathbb Q$ be the splitting field of $f(X)=X^n-X-1$. The Osada polynomial-family conjecture. For every root $u$ of $f$, the corresponding $L_f$-rational point is non-torsion in $J_C(L_f)$ and

$$\dim J_C(L_f)\otimes\mathbb Q\geq m(\operatorname{Gal}(L_f/\mathbb Q))=n-1.$$

This is presented as a particular case of the preceding primitive-point conjecture; the source supplies no resolution.