Growth of direct products

Question

Let $G$ and $H$ be finitely generated groups. Is it true that $ \gamma_{G \times H} $ is weakly equivalent to $ \gamma_{G}\gamma_{H} $?

Assuming the answer is yes, can one use exactly the same proof as in this post? (If $S$ and $T$ are the sets of generators of $G$ and $H$, I take $S\times 1 \cup 1 \times T$ as a generating set for $G \times H$)

Answer 1

Yes, since $$ \gamma_{G\times H}(r) \;\leq\; \gamma_G(r)\, \gamma_H(r) $$ and $$ \gamma_G(r)\,\gamma_H(r) \;\leq\; \gamma_{G\times H}(2r). $$