Equiconsistency of $\textbackslash \kappa$-compactness and $\textbackslash Pi^1_1$-subcompactness

A cardinal $\textbackslash \kappa$ is $\textbackslash \kappa$-compact when it has the compactness property described in the surrounding discussion, and a cardinal $\textbackslash \delta$ is $\textbackslash \delta^+$-$\textbackslash Pi^1_1$-subcompact when it has the corresponding elementary-embedding property. The claim concerns the existence of cardinals with these properties.

Equiconsistency conjecture. The existence of $\textbackslash \kappa$ which is $\textbackslash \kappa$-compact is equiconsistent with the existence of a cardinal $\textbackslash \delta$ which is $\textbackslash \delta^+$-$\textbackslash Pi^1_1$-subcompact.

The conjecture is motivated by reflection results connecting strong compactness, compactness, and $\textbackslash Pi^1_1$-subcompactness, together with inner-model consequences from failures of square. Its resolution is not supplied in the source.