*I'm sadly an outsider to nonclassical propositional logics. All terminology below comes from Humberstone's book The Connectives, specifically section 4.2.*

A new connective - a bit more precisely, a set of rules in some appropriate form for said connective - is **conservative** over a deductive system $\mathcal{L}$ if adjoining it does not result in any new valid sequents in the original language of $\mathcal{L}$ itself. The standard non-example of this with $\mathcal{L}$ = classical propositional logic is Prior's $\mathsf{tonk}$, with rule set $$A\vee B\vdash A\mathsf{\,tonk\,}B\quad\mbox{and}\quad A\mathsf{\,tonk\,}B\vdash A\wedge B.$$

Interestingly, two individually-conservative connectives may not be jointly conservative. Elaborating on C.G. McKay 1985, Humberstone observes that the connectives $O$ and $\perp$ described by the schemes$$(A\rightarrow OA)\rightarrow A\quad\mbox{and}\quad \perp\rightarrow A$$ are separately-but-not-jointly conservative over positive logic (= the fragment of propositional intuitionistic logic using only $\wedge,\vee,\rightarrow$).

This suggests the following. Say that a connective $\star$ (or more accurately again, a set of appropriate rules for same) is **hereditarily conservative** over $\mathcal{L}$ iff $\star$ and $@$ are jointly conservative over $\mathcal{L}$ whenever $@$ is conservative over $\mathcal{L}$. Obviously any "already-definable" connective is hereditarily conservative, but beyond this not much is clear to me. I'm interested in any information about hereditary conservativity, but to get things off the ground here's a concrete question:

Are there any hereditarily conservative connectives over positive logic which are not already definable over positive logic?

*(I could not find an answer to this question in Humberstone, but that book is large enough that I'm not confident in my searching abilities. Meanwhile, the "universal-algebra" tag is admittedly tenuous but there is apparently no "nonclassical-logic" or "propositional-logic" tag, so it seems the best available.)*