Finite powers of small strong measure zero sets

Let $X$ be a set of reals with strong measure zero, and suppose that $|X|<\mathfrak{b}$, where $\mathfrak{b}$ is the bounding number. A finite power of $X$ is a Cartesian product $X^n$ for some positive integer $n$. Finite-power strong measure zero conjecture. Every finite power $X^n$ has strong measure zero; equivalently, every finite power satisfies

$$\mathsf{S}_1(\mathcal{O},\mathcal{O}).$$

This would provide the finite-power property needed in the surrounding discussion of the Gerlitz–Nagy covering property and Borel Conjectures. The source presents this as a statement that needs to be proved, and no resolution is supplied here.