  We show that the existence of a well-known type of ideals on a regular
cardinal $\lambda$ implies a compactness property concerning the
specialisability of a tree of height $\lambda$ with no cofinal branches. We
also use Neeman's method of side conditions to show that the existence of such
ideals is consistent with stationarily many appropriate guessing models. These
objects suffice to extend the main theorem of \cite{mhpr_spe}: one can
generically specialise any branchless tree of height $\kappa^{++}$ with a
${<}\kappa$-closed, $\kappa^{+}$-proper, and $\kappa^{++}$-preserving forcing,
which has the $\kappa^+$-approximation property.
