TITLE: Questions about $\aleph_1-$closed forcing notions
QUESTION [15 upvotes]: "Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including:
1) $GCH$ fails everywhere,
2) there are no inaccessible cardinals,
3) there are no $\kappa-$Souslin trees,
4) Any non-trivial $c.c.c.$ forcing adds a real,
5) Any non-trivial $\kappa^+-$closed forcing notion collapses some cardinals.
Consistency of (1) is proved  by Foreman-Woodin, (2) clearly can be consistent and the consistency of (4) is shown in "Forcing with c.c.c forcing notions, Cohen reals and Random reals".
My interest is in the consistency of (5). Let's consider the case $\kappa=\omega.$
Question 1. Is it consistent that any non-trivial $\aleph_1-$closed forcing notion collapses some cardinals? 
The above question seems very difficult, and it is not difficult to show that for its consistency we need some very large cardinals. But maybe the following is simpler:
Question 2. Is it consistent that any non-trivial $\aleph_1-$closed forcing notion of size continuum collapses some cardinals? Does its consistency imply the existence of large cardinals?

REPLY [3 votes]: The following results are obtained in a joint work with Yair Hayut and are now presented in our joint paper On Foreman's maximality principle:

Theorem 1. (Assuming the existence of a strong cardinal) There is a model of $ZFC$ in which for all uncountable (regular) cardinals $\kappa,$ any $\kappa$-closed forcing notion of size $\leq 2^{<\kappa}$ collapses some cardinals.

Note that in such a model $GCH$ should fail everywhere and hence some very large cardinals are needed. The following theorem is in the opposite direction:

Theorem 2 (Assuming the existence of a strong cardinal and infinitely many inaccessibles above it) There is a model of $ZFC$ in which $GCH$ fails everywhere and for each uncountable (regular) cardinal $\kappa,$ there exists a $\kappa$-closed forcing notion of size $2^{<\kappa}$ which preserves all cardinals.