Full Souslin trees at small cardinals

A kappa$\kappa$-tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full kappa$\kappa$-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal kappa$\kappa $. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be aleph 3$\aleph _3$ many full aleph 2$\aleph _2$-trees such that the product of any countably many of them is an aleph 2$\aleph _2$-Souslin tree.