This should be a consequence of classification of injective quasi-coherent $\mathcal{O}_X$-modules over (locally) Noetherian schemes:

By Theorem II.7.17 of Hartshorne's Residues and Duality (cf. B. Conrad's Grothendieck Duality and Base change, Lemma 2.1.5), every injective quasi-coherent sheaf on $X$ is of the form $\bigoplus_i \mathscr{J}(x_i),$ for some collection of (possibly repeating) points $\{x_i \;|\; i \in I\}$, where $\mathscr{J}(x_i)=\nu_*(\widetilde{ E(k(x_i))}),$ $\nu: \mathrm{Spec}\,\mathcal{O}_{X, x_i} \rightarrow X$ is the canonical map and $E(k(x_i))$ is the $\mathcal{O}_{X, x_i}$-injective hull of the residue field $k(x_i)$.

In the case when $X$ is integral, the torsion-free assumption clearly excludes contribution of any point $x_i \in X$ apart from the generic point (note that $\mathscr{J}(x_i)$ has support $\overline{\{x_i\}}$).

Thus, a torsion-free injective quasi-coherent sheaf on an integral scheme is of the form $\mathscr{J}(\eta)^{\oplus I}$ for some set $I$, where $\eta$ is the generic point of $X$. But it is easy to see that $\mathscr{J}(\eta)=\nu_*(\widetilde{E(k(\eta))})=\nu_*(\widetilde{\mathcal{O}_{X, \eta}})=\mathcal{K}_X$.