Let $X$ be a smooth curve over a field. Let $Y$ be the triple product $X \times X \times X$. Let $\gamma$ be a homologically trivial codimension $2$ cycle.

In the text [Zhang, p. 76] that I am currently reading it is concluded that $\pi_{i,*}(\gamma)$ is trivial (rationally). I do not see why this is true, and could not find a proof elsewhere.

In [Zhang] there is more context, but I think that I stated the relevant input for the claim.

Reference

[Zhang] Shou-Wu Zhang. “Gross–Schoen Cycles and Dualising Sheaves”. arXiv: http://arxiv.org/abs/0812.0371 .

(I should add a note that I am currently not in the situation to search through a library. So maybe this can be found in any book on algebraic cycles; if so, please give the reference.)