The restricted-product conjecture for quasi-endomorphisms of localized integers

Let $S$ be a multiplicatively closed subset of $\mathbb Z$ and let $\hat{S}$ denote its saturation. Suppose that $\hat{S}$ is generated by a possibly infinite number of primes $p_1,p_2,\ldots$.

Restricted-product conjecture. The ring of quasi-endomorphisms satisfies

$$\operatorname{QEnd}\left(S^{-1}\mathbb Z/\mathbb Z\right)\cong\sideset{}{'}\prod_{p_i}(\mathbb Q_{p_i},\mathbb Z_{p_i}).$$

This would give a complete characterization of the ring of quasi-homomorphisms of $S^{-1}\mathbb Z/\mathbb Z$ and extends the finite-prime calculation to an arbitrary possibly infinite set of primes. The source gives no resolution of this conjecture.