Abstract : We report on a joint work in progress with Rahman Mohammadpour in which we study the problem of the possible existence of a universal tree under weak embeddings in the classes of $\aleph_{2}$-Aronszajn and wide $\aleph_{2}$-Aronszajn trees. This problem is more complex than previously thought, in particular it seems not to be resolved under ShFA $+$ CH using the technology of weakly Lipshitz trees. We show that under CH, for a given $\aleph_{2}$-Aronszajn tree $\mathrm{T}$ without a weak ascent path, there is an $\aleph_{2^{-\mathrm{C}\mathrm{C}}}$ countably closed forcing forcing which specialises $\mathrm{T}$ and adds an $\aleph_{2}$-Aronszajn tree which does not embed into T. One cannot however apply the ShFA to this forcing.

Keywords : $\omega_2$-Aronszajn tree; universality