On associative algebras satisfying the identity $x^5=0$

We study Kuzmin's conjecture on the index of nilpotency for the variety ${\mathcal {N}il}_5$ of associative nil-algebras of degree 5. Due to Vaughan–Lee [11] the problem is reduced to that for $k$-generator ${\mathcal {N}il}_5$-superalgebras, where $k\leq 5$. We confirm Kuzmin's conjecture for 2-generator superalgebras proving that they are nilpotent of degree 15.