Kuzmin's conjecture on the nilpotency class of nil algebras

Let $N(n)$ denote the least integer such that every nonunitary nil algebra over a field of characteristic $0$ with $r^n=0$ for all elements $r$ satisfies $r_1\cdots r_{N(n)}=0$ for all choices of elements $r_1,\ldots,r_{N(n)}$. The known bounds are

$$\frac{n(n+1)}{2}\leq N(n)\leq n^2.$$

Kuzmin's conjecture. The exact value of the nilpotency class is

$$N(n)=\frac{n(n+1)}{2}.$$

The conjecture seeks the sharp value in the characteristic-zero Nagata–Higman theorem. Kuzmin established the lower bound, while the best known general upper bound is $N(n)\leq n^2$; equality of the two bounds is not established in the supplied text.