The implication from positive-set separation to submodule separation

Let Conjecture c:one be the assertion that pure states separate the localized image of an $\sA$-submodule from a point outside it, and let Conjecture c:two be the assertion that every closed $\sA$-convex subset $A\subset\sA_+$ avoiding $0$ admits a pure state uniformly positive on $A$. The implication conjecture. Conjecture c:two implies Conjecture c:one. This is presented as a logical reduction: establishing the positive-set separation statement would establish the pure-state separation statement for submodules. The source does not claim that either conjecture has been proved.