Souslin trees which are hard to specialise

We construct some $\kappa ^+$-Souslin trees which cannot be specialised by any forcing which preserves cardinals and cofinalities. For $\kappa$ a regular cardinal we use the $\begin {picture}(6,6)(0,0) \drawline (0,3)(3,6)(6,3) (3,0)(0,3) \drawline (0,6)(6,6)(6,0) (0,0)(0,6) \end {picture}$ principle, and for $\kappa$ singular we use squares and diamonds.