TITLE: Consistency strength of $\aleph_2$-Souslin hypothesis
QUESTION [15 upvotes]: Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis?
Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and $\aleph_3$-Souslin hypothesis?
Remark 1. By $\kappa$-Souslin hopothesis, I mean there are no $\kappa$-Souslin trees.
Remark 2. By Laver-Shelah, the existence of a weakly compact cardinal implies the consistency of  $\aleph_2$-Souslin hypothesis. On the other hand by results of Shelah-Stanly, if we assume some instances of $GCH$+ $\aleph_2$-Souslin hypothesis (having $CH$ is sufficient), then some large cardinals (at least Mahlo) are required. In the above question I do not take care of preserving instances of $GCH$.

REPLY [11 votes]: Answer to 1, without CH:

Mitchell and Silver, 1973: Weakly compact is an upper bound.

Answer to 1, with CH:

Laver and Shelah, 1981: Weakly compact is an upper bound.
Shelah and Stanley, 1982: Inaccessible is a lower bound.

Answer to 1, with GCH:

Gregory, 1976: Mahlo cardinal is a lower bound.
Rinot, 2016: Weakly compact is a lower bound.

Answer to 2, with GCH: 

Rinot, 2016 (building on recent work of Schindler and Steel): AD
holds in $L(\mathbb R)$ is a lower bound.