Avoiding 2-binomial squares and cubes

Two finite words $u,v$ are $2$-binomially equivalent if, for all words $x$ of length at most $2$, the number of occurrences of $x$ as a (scattered) subword of $u$ is equal to the number of occurrences of $x$ in $v$. This notion is a refinement of the usual abelian equivalence. A $2$-binomial square is a word $uv$ where $u$ and $v$ are $2$-binomially equivalent.

In this paper, considering pure morphic words, we prove that $2$-binomial squares (resp. cubes) are avoidable over a $3$-letter (resp. $2$-letter) alphabet. The sizes of the alphabets are optimal.