On compactness of weak square at singulars of uncountable cofinality

Cummings, Foreman, and Magidor proved that Jensen's square principle is non-compact at $\aleph_\omega$, meaning that it is consistent that $\square_{\aleph_n}$ holds for all $n<\omega$ while $\square_{\aleph_\omega}$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild hypotheses, the weak square principle $\square_\kappa^*$ is in fact compact at singulars of uncountable cofinality, and that an even stronger version of these hypotheses is not enough for compactness of weak square at $\aleph_\omega$.