  In this note we study the behavior of symplectic capacities of convex domains
in the classical phase space with respect to symplectic $p$-products. As an
application, by using a "tensor power trick", we show that it is enough to
prove the weak version of Viterbo's volume-capacity conjecture in the
asymptotic regime, i.e., when the dimension is sent to infinity. In addition,
we introduce a conjecture about higher-order capacities of $p$-products, and
show that if it holds, then there are no non-trivial $p$-decompositions of the
symplectic ball.
