Aronszajn trees and maximality

Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal $\mu$, we prove the consistency of the existence of a wide ${\mu }^{+}$-Aronszajn tree, i.e. a tree of height and cardinality ${\mu }^{+}$ with no branches of length ${\mu }^{+}$, into which every wide ${\mu }^{+}$-Aronszajn tree can be embedded.