It is a long-standing open problem whether the Sierpiński carpet attains its conformal dimension or not. While this problem remains unresolved, we prove that Cartesian products $\mathbb{S}^k$, where $\mathbb{S}$ is the Sierpiński carpet and $k \geq 2$, do not attain their conformal dimension. Our approach is based on the Sobolev spaces and energy measures on $\mathbb{S}$ - constructed by Shimizu, Kigami, and Murugan and Shimizu - together with a certain singularity result of energy measures from the theory of analysis on fractals.