Using the lottery preparation, we prove that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all ${\lt}\kappa$-closed + $\kappa^+$-preserving forcing. This degree of indestructibility, we prove, is the best possible from this hypothesis within the class of ${\lt}\kappa$-closed forcing. From a stronger hypothesis, however, we prove that the strong unfoldability of $\kappa$ can be made indestructible by all ${\lt}\kappa$-closed forcing. Such indestructibility, we prove, does not follow from indestructibility merely by ${\lt}\kappa$-directed closed forcing. Finally, we obtain global and universal forms of indestructibility for strong unfoldability, finding the exact consistency strength of universal indestructibility for strong unfoldability.