Reconstruction theorem for monoid schemes

We aim to reconstruct a monoid scheme $X$ from the category of quasi-coherent sheaves over it. This is much in the vein of Gabriel's original reconstruction theorem. Under some finiteness condition on a monoid schemes $X$, we show that the localising subcategories of the topos $\mathfrak{Qc}(X)$ of quasi-coherent sheaves on $X$ is in a one-to-one correspondence with open subsets of $X$, while the elements of $X$ correspond to the topos points of $\mathfrak{Qc}(X)$. This allows us to reconstruct $X$ from $\mathfrak{Qc}(X)$.