Does the following limit exist? $$ \lim_{n \to \infty}\, n^{-3/2} \min_{a_{ij}=\pm 1}\max_{x_{j}=\pm 1}\left|\sum_{1\leq i <j \leq n} a_{ij}x_{i}x_{j} \right| $$ There is no any significant motivation behind this question, it is just a pure curiosity.

I am not really interested in numerical computations, and proofs of upper or lower bounds unless they provide significant evidence that the limit does not exist. In case the limit exists I am curious what is the proof?