Can $\square_{\kappa,\kappa}$ fail everywhere?

Question

A theorem in Cummings-Magidor, Martin's Maximum and weak square:

Theorem. Assume MM. Then we have, for $\lambda>\omega$,

1. $\square_{\omega_1,\omega_1}$ fails.
2. If $\mathrm{cf}(\lambda)=\omega$, then $\square_{\lambda,\lambda}$ fails.
3. If $\mathrm{cf}(\lambda)=\omega_1<\lambda$, then $\square_{\lambda,<\lambda}$ fails.
4. If $\mathrm{cf}(\lambda)>\omega_1$, then $\square_{\lambda,<\mathrm{cf}(\lambda)}$ fails.

(In fact, in 2, even $\mathrm{Apr}_\lambda$ fails, but I haven't found the reference yet.)
Moreover, the theorem is also optimal in the following sense: It is consistent with MM that

1. $\square_{\lambda,\lambda}$ holds for all $\lambda$ with $\mathrm{cf}(\lambda)=\omega_1<\lambda$,
2. $\square_{\lambda,\mathrm{cf}(\lambda)}$ holds for all $\lambda$ with $\mathrm{cf}(\lambda)>\omega_1$.

I think this is the strongest negation of (Jensen's) squares that we know. If not then please let me know.

$\square_{\kappa,\kappa}$ is equivalent to existence of special $\kappa^+$-Aronszajn tree. Therefore the question is equivalent to asking whether it is consistent that if $\kappa^+$ is a successor cardinal then there is no special $\kappa^+$-Aronszajn tree.

Answer 1

This is open. The best result seems to be the following theorem of Spencer Unger:

Theorem From large cardinals, it is consistent that there are no special Aronszajn trees at all regular cardinals in the interval $[\aleph_2, \aleph_{\omega^2+3}]$.

See his papers SUCCESSIVE FAILURES OF APPROACHABILITY.