The higher-limit map for abelian-group presentations

Let $A$ be an abelian group, and let ${\sf Pres}_{\sf Ab}(A)$ be the category of presentations $H\rightarrowtail F\twoheadrightarrow A$. Define the functor

$$\mathfrak F:{\sf Pres}_{\sf Ab}(A)\to \sf Ab$$

by

$$\mathfrak F(\{H\rightarrowtail F\twoheadrightarrow A\})=H.$$

The preceding natural monomorphism

$$A\hookrightarrow\,\varprojlim{}^1 \mathfrak F$$

Higher-limit conjecture. The above map is a natural isomorphism.

This asserts that the first higher inverse limit of the kernel functor on abelian-group presentations recovers the presented abelian group itself. The supplied text gives the natural monomorphism but does not provide evidence that the asserted isomorphism is known or resolved.