The Structural Reflection characterization of least large cardinals

Let $\kappa$ be a cardinal. A cardinal is weakly inaccessible when it has the corresponding weak inaccessibility property in some inner model. An SRP is a Structural Reflection Principle, and a large-cardinal notion is a property defining a class of large cardinals.

Least-cardinal structural reflection conjecture. A cardinal $\kappa$ is the least cardinal satisfying some large-cardinal notion if and only if $\kappa$ is the least cardinal satisfying some Structural Reflection Principle that implies, in some inner model, that $\kappa$ is weakly inaccessible.

This is presented as a further conjecture intended to give necessary and sufficient conditions for a cardinal to count as large. The source gives no resolution status.