The product-space conjecture for D-separability

Let $X$ be a topological space, and let $d(X)$ denote its density and $d(X)^+$ its successor cardinal. A space is D-separable if every sequence of dense subsets admits a choice of one point from each set such that the set of chosen points is dense.

Product-space conjecture. The space

$$X^{d(X)^+}$$

is never D-separable.

This is presented as an open problem following the theorem that $X^{\omega_1}$ is not D-separable for every separable space $X$ with at least two points, and that $\mathfrak{ds}=\omega_1$. The conjecture asks whether the exponent can generally be reduced to the successor of the density.