Recall that the *square partition relation* $\kappa\to[\lambda]^k_\eta$ holds iff for every $f:[\kappa]^k\to\eta$ there exists $H\in[\kappa]^\lambda$ such that $f"[H]^k\neq\eta$. I.e. said in words, everytime we colour the $k$-element subsets of $\kappa$ in $\eta$ many colours, we can find a subset $H\subseteq\kappa$ of size $\lambda$ such that we omit a colour when colouring $k$-element subsets of $H$.

Note that for $\kappa>\omega$, $\kappa\to[\kappa]^2_2$ holds iff $\kappa$ is weakly compact, and it's not too hard to see that $\kappa\to[\kappa]^2_2$ implies $\kappa\to[\kappa]^2_\kappa$. Furthermore, it's a result of Rinot ('14) that every regular uncountable cardinal $\kappa$ satisfying $\kappa\to[\kappa]^2_\kappa$ is threadable, which by a result of Todorcevic ('87) implies that such a cardinal is weakly compact in $L$, making the two notions equiconsistent. This leads me to my question.

Question.Can we separate inaccessible cardinals $\kappa$ satisfying $\kappa\to[\kappa]^2_\kappa$ from the weakly compacts? By this I mean showing that $\textsf{ZFC}+\exists\text{weakly compact}$ can't prove that every inaccessible $\kappa$ satisfying $\kappa\to[\kappa]^2_\kappa$ is weakly compact.

The standard way to do this is to use Kunen's method of starting with a weakly compact cardinal, adding a $\kappa$-Souslin tree to kill the weak compactness, resurrect the weak compactness in a further forcing extension and show that $\kappa$ has the desired property in the intermediate extension. But the presence of such a tree either makes $\kappa$ singular or implies that $\kappa\not\to[\kappa]^2_\kappa$ (this is mentioned on slide 12 of Rinot's ESTC '17 talk), so that doesn't work in this case.

**EDIT:** This edit is based on the comments below. Replaced Todorcevic' result that regular uncountable cardinals $\kappa$ satisfying $\kappa\to[\kappa]_\kappa^2$ reflects stationary sets to Rinot's result that such cardinals are threadable. Both are true, but we need threadability to show the equiconsistency.