Identities equivalent to the reflexive identity

Question

This is a follow-up to my previous question, here: A characterization of the identities which are equivalent to the trivial identity. As in that question, let our signature be that of a single binary operation $+$. The reflexive identity is $x=x$. I conjecture that the only identities equivalent to the reflexive identity are those of the form $t=t$, for some term $t$. Is this true?

Answer 1

It is true.

If $t(x_1,\dots,x_n)$ and $s(x_1,\dots,x_n)$ are different terms in no more than $n$ variables, then in the term algebra of one binary operation, $t = s$ doesn't hold.
So it can't be equivalent to $x=x$ which holds in every algebra (in particular, in the term algebra).