Higher-dimensional inhomogeneous Hausdorff-measure divergence conjecture

Let $m,d\geq1$ be integers, let $\Psi:\mathbb Z^m\to(0,\infty)$, and let $\WW^{md}_{\Psi,\bftheta}\subset\I^{md}$ be the associated inhomogeneous multiplicative approximation set. Let $f$ be a dimension function satisfying the source's stated hypotheses for some $s\in(md-1,md)$ and $C>0$. Define

$$\sum_{\mathbf q\in\mathbb Z^m}|\mathbf q|^{md}\Psi^{-md+1}(\mathbf q)f\left(\frac{\Psi(\mathbf q)}{|\mathbf q|}\right).$$

Higher-dimensional Hausdorff-measure divergence conjecture. If $\Psi(\mathbf q)=\psi(|\mathbf q|)$ for a monotonically decreasing $\psi:\mathbb N\to(0,\infty)$, this series diverges, and $x\mapsto x^{-md+1}f(x)$ is monotonically increasing, then

$$\HH^f(\WW^{md}_{\Psi,\bftheta}\cap\I^{md})=\infty.$$

The paper gives the corresponding convergence implication as a zero-measure result and explains that the divergence claim does not follow directly from the Slicing Lemma. It is therefore posed as an open complementary problem.