Conjecture on properness and incomparability of positive free logic systems

The paper considers the classical positive free logic systems appearing in Theorem

, namely the systems ordered there by inclusion, together with $\mathbf{CPF}^{\begin{sideways}\begin{sideways}$\iota$\end{sideways}\end{sideways} H}$, $\mathbf{CPF}^{\begin{sideways}\begin{sideways}$\iota$\end{sideways}\end{sideways} S}$, and $\mathbf{FD2}$. **Conjecture on system inclusions.** (a) All inclusions of Theorem

are proper. (b) \mathbf{CPF}^{\begin{sideways}\begin{sideways}\iota\end{sideways}\end{sideways} H} and \mathbf{CPF}^{\begin{sideways}\begin{sideways}\iota\end{sideways}\end{sideways} S} are incomparable. (c) \mathbf{CPF}^{\begin{sideways}\begin{sideways}\iota\end{sideways}\end{sideways} H} and $\mathbf{FD2}$ are incomparable. This conjecture concerns the exact relationships among the positive free logic systems; the supplied text gives no evidence that these properness and incomparability claims have been resolved.