Using the lottery preparation, we prove that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all $<\kappa$-closed + ${\kappa }^{+}$-preserving forcing. This degree of indestructibility, we prove, is the best possible from this hypothesis within the class of $<\kappa$-closed forcing. From a stronger hypothesis, however, we prove that the strong unfoldability of $\kappa$ can be made indestructible by all $<\kappa$-closed forcing. Such indestructibility, we prove, does not follow from indestructibility merely by $<\kappa$-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.