The famous Amitsur-Levitzki Theorem states that the algebra $M_n(\mathbb C)$ satisfies no polynomial identity of degree less than $2n$ and it satisfies $$p(x_1,\dots,x_{2n}) = \sum_{\sigma \in \frak S_{2n} }{\rm sign} (\sigma) \cdot x_{\sigma(1)} \cdots x_{\sigma(2n)},$$ a polynomial in $2n$ non-commuting variables.

Question: What is the smallest degree of a non-trivial polynomial identity of $M_n(\mathbb C)$ in two variables?

By inserting non-commutative monomials in two variables into the Amitsur-Levitzki polynomial one gets an upper abound of the degree of $$2n \lceil \log_2(2n) \rceil.$$ Is $O(n \log(n))$ optimal? The question might be approachable for low $n$, but I am more interested in asymptotic properties.