If $R$ and $S$ are complete Huber rings with $varphi: R to S$ a continuous map, then is it true in general that if $mathrm{Spa}(S, S^circ) to mathrm{Spa}(R, R^circ)$ is an open immersion of adic spaces (here $S^circ$ and $R^circ$ are the power-bounded subrings) then $mathrm{Spec}(S) to mathrm{Spec}(R)$ is injective?

For example, this is true if $R$ and $S$ both have the discrete topology, because if $frak p$ and $frak q$ are two prime ideals in $S$ which are equal after restricting to $R$ then $(frak p, |cdot|_{rm triv})$ and $(frak q, |cdot|_{rm triv})$ (trivial valuations), which are both points in $mathrm{Spa}(S,S)$, restrict to the trivial valuation on $R/varphi^{-1}(frak p)$.

But I’m not sure how generally to expect that this is true.