Razmyslov-type upper-bound conjecture for nilpotency degrees

Let $N_{n,d}$ be the relatively free associative algebra with identity $x^n=0$, and let $C_{n,d}$ be its nilpotency degree. Assume $n,d\geq2$ and that the characteristic $p$ of the infinite base field satisfies $p>n$.

Upper-bound conjecture.

$$C_{n,d}\leq n^2.$$

This generalizes Razmyslov's upper bound from characteristic zero to the case $p>n$. The assertion is known for $n=2,3$, while the general case is left as a conjecture.