A wide Aronszajn tree is a tree of size and height $\omega_1$ with no uncountable branches. We prove that under $MA(\omega_1)$ there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and
V\"a\"an\"anen from 1994.

We also prove that under $MA(\omega_1)$, every wide Aronszajn tree weakly embeds in an Aronszajn tree, which combined with a result of Todor{\v c}evi{\'c} from 2007, gives that under $MA(\omega_1)$ every wide Aronszajn tree embeds into a Lipschitz tree or a coherent tree. We also prove that under $MA(\omega_1)$ there is no wide Aronszajn tree which weakly embeds all Aronszajn trees, improving the result in the first paragraph as well as a result of Todor{\v c}evi{\'c} from 2007 who proved that under $MA(\omega_1)$ there are no universal Aronszajn trees.