TITLE: Consistency strength of being strong cardinal and indestructible under collapses
QUESTION [8 upvotes]: What is the consistency strength of the following statement:

$\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$  where $\theta> \kappa$ is some fixed inaccessible cardinal.

Remark 1. By Laver's result, the above statement is consistent relative to the existence of a supercompact cardinal (and an inaccessible above it).
Remark 2. By results of Mitchell, Schimmerling and Steel, ``the covering lemma up to a Woodin cardinal'', if $\kappa$ remains measurable after collapsing $\kappa^+$ to $\kappa,$
then there is an inner model with a Woodin cardinal.
Remark 3. Apter's paper ``Strong cardinals can be fully indestructible'' have some positive consistency results related to the question.

REPLY [4 votes]: If κ is weakly compact in VCol(κ,κ+), then there is a non-domestic premouse, see Theorem 0.4 of  https://ivv5hpp.uni-muenster.de/u/rds/stacking_mice.pdf