Equational identities equivalent to the associative identity

Question

This is a natural follow-up to my previous question, here: Is only the commutative identity equivalent to the commutative identity?. As usual, let our signature be that of a single binary operation $+$. Suppose $E$ is an equational identity which is equivalent to the associative identity $(x+y)+z=x+(y+z)$. Must $E$ be an alphabetical variant of the associative identity? That is, must $E$ be of the form $(v_1 + v_2) + v_3 = v_1 + (v_2 + v_3)$, or its reverse $v_1 + (v_2 + v_3) = (v_1 + v_2) + v_3$, for three distinct variables $v_1, v_2, v_3$?