Square with Built-in Diamond-plus

We formulate combinatorial principles that combine the square principle with various strong forms of the diamond principle, and prove that the strongest amongst them holds in L for every infinite cardinal. As an application, we prove that the following two hold in L: (1) For every infinite regular cardinal λ, there exists a special λ-Aronszajn tree whose projection is almost Souslin; (2) For every infinite cardinal λ, there exists a respecting λ-Kurepa tree; Roughly speaking, this means that this λ-Kurepa tree looks very much like the λ-Souslin trees that Jensen constructed in L.