The weakly compact reflection principle need not imply a high order of weak compactness

The weakly compact reflection principle $\text{Refl}_{\text{wc}}(\kappa)$ states that $\kappa$ is a weakly compact cardinal and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. The weakly compact reflection principle at $\kappa$ implies that $\kappa$ is an $\omega$-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that $\kappa$ is $(\omega+1)$-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at $\kappa$ then there is a forcing extension preserving this in which $\kappa$ is the least $\omega$-weakly compact cardinal. Along the way we generalize the well-known result which states that if $\kappa$ is a regular cardinal then in any forcing extension by $\kappa$-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $\kappa$ is a weakly compact cardinal then after forcing with a `typical' Easton-support iteration of length $\kappa$ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.