On the word problem for the free Burnside semigroups satisfying $x^2=x^3$

We study the word problem for free Burnside semigroups satisfying the identity $x^2=x^3$. For any $k>2$ we prove that the word problem for the $k$-generated free Burnside semigroup $B(2,1,k)$ can be reduced to the word problem for the two-generated semigroup $B(2,1,2)$. Moreover, if every element of $B(2,1,2)$ is a regular language, then every element of $B(2,1,k)$ also appears to be a regular language. Therefore, the semigroup $B(2,1,k)$ satisfies the Brzozowski conjecture if and only if so does $B(2,1,2)$.