We continue our research on the relative strength ofL-like combinatorial principles for successors of singular cardinals. In [3] we have shown that the existence of a λ+-special Aronszajn tree does not follow from that of a λ+-Souslin tree. It follows from [5], [4] and [6] that under G.C.H. □λ does imply the existence of a λ+-Souslin tree. In [2] we show that □λ does not follow from the existence of a λ+-special Aronszajn tree. Here we show that the existence of such a tree implies that of an ‘almost Souslin’ λ+-tree. It follows that the statement “All λ+-Aronszajn trees are special” implies that there are no λ+-Aronszajn trees.