A famous question in symplectic topology, Viterbo’s volume-capacity conjecture [C. Viterbo, J. Am. Math. Soc. 13, No. 2, 411–431 (2000; Zbl 0964.53050)], states that among all convex bodies in the Euclidean space of dimension \(2n\) with a given volume, the symplectic capacity is maximal for symplectic images of the \(2n\)-dimensional Euclidean ball. The paper studies the behavior of the Ekeland-Hofer-Zehnder symplectic capacity of convex domains in the classical phase space with respect to symplectic \(p\)-products, using a “tensor power trick”. The authors obtain as a corollary that the proof of the conjecture for the Ekeland-Hofer-Zehnder symplectic capacity can be reduced to the case when the dimension is sent to infinity. This very nice result probably opens up a way to proving the weak version of Viterbo’s volume-capacity conjecture. Moreover, a conjecture about higher-order Ekeland-Hofer capacities of symplectic \(p\)-products is introduced, and it is shown that for \(p\ge 1\) and \(p\ne 2\) the symplectic \(p\)-product of any two convex bodies is not symplectomorphic to a Euclidean ball if this conjecture holds. The authors prove a corresponding result on higher-order Gutt-Hutchings capacities convex (or concave) toric domains to support the conjecture.