Let $R$ be an integral domain and $M$ be a torsion free injective module. Then $M$ is of the form $\oplus_{I}K$ (for some index set $I$), where $K$ is the quotient field of $R$.

Now let $\cal F$ be a torsion free injective quasi coherent sheaf over an integral scheme, and ${\cal K}_X$ denote the quasi-coherent sheaf determined by $U\mapsto \rm Frac({\cal O}_X(U))$. Then :

Is any torsion free qusai-coherent injective sheaf of the form $\oplus_{I}{\cal K}_X$ for some index set $I$?