The question in the title is somewhat self contained but let me make some definitions and remarks to clarify.

Recall that $\mathsf{MA}^+(\sigma\text{-closed})$ is the statement that if $\mathbb P$ is a $\sigma$-closed poset, $\langle D_\alpha \; |\; \alpha < \omega_1\rangle$ is a sequence of dense subsets of $\mathbb P$ and $\dot{S}$ is a name for a stationary set then there is a filter $G \subseteq \mathbb P$ so that $G \cap D_\alpha \neq \emptyset$ for all $\alpha < \omega_1$ and $\{\alpha \; | \; \exists p \in G \; p \Vdash \check{\alpha} \in \dot{S}\}$ is stationary.

A Kurepa tree is a tree of height $\omega_1$, countable levels and $\geq \omega_2$ branches.

My question is simply if $\mathsf{MA}^+(\sigma\text{-closed})$ implies there are no Kurepa trees.

I have no intuition about this, as on the one hand I don't see a way that $\sigma$-closed forcing suffices to show there are no Kurepa trees and on the other hand I don't see a way to force $\mathsf{MA}^+(\sigma\text{-closed})$ without killing all Kurepa trees. Specifically:

It's well known that Silver collapsing an inaccessible to $\omega_2$ implies that there are no Kurepa trees so in natural models of $MA^+(\sigma\text{-closed})$ obtained by iterating all $\sigma$-closed forcing notions below some sufficiently large inaccessible (supercompact?) $\kappa$ there are no Kurepa Trees.

Meanwhile, the proof that $\mathsf{PFA}$ implies there are no Kurepa trees involves specializing an Aronszajn tree and such specialization forcings are not in general $\sigma$-closed.

Thanks!