Conjectured Hausdorff-measure zero-one law for algebraic points near manifolds

Let $\bm f:\operatorname{dom}(\bm f)\subseteq\mathbb R^d\to\mathbb R^{m+1}$ be the map in the source, and let $\psi_0,\ldots,\psi_m:\mathbb R^+\to\mathbb R^+$ be decreasing functions satisfying $\psi_i\in O(\psi_j)$ for $0\leq i<d$ and $d\leq j\leq m$. Suppose that for some $\kappa>0$,

$$\kappa^{n-m+d}>\lim_{t\to\infty}\frac{\psi_d(\kappa^t)\cdots\psi_m(\kappa^t)}{\psi_d(\kappa^{t+1})\cdots\psi_m(\kappa^{t+1})}.$$

Let $g$ be a dimension function with $r^{-d}g(r)$ non-increasing, and assume that $\bm f$ is Lipschitz continuous, $\operatorname{Vand}(\bm f)\neq0$, $\det(U^0_{\bm f}\cdots U^m_{\bm f})\neq0$, and satisfies condition~. Define

$$S_{n,d}^g(\psi_0,\ldots,\psi_m)=\sum_{Q=1}^{\infty}Q^n\frac{\psi_d(Q)\cdots\psi_m(Q)}{Q^{m+1-d}}g\left(\frac{\psi_0(Q)\cdots\psi_{d-1}(Q)}{Q^d}\right).$$

Manifold Khintchine-type conjecture. The Hausdorff $g$-measure of $\dualappr_{n,\bm f}^*(\psi_0,\ldots,\psi_m)$ is zero if $S_{n,d}^g(\psi_0,\ldots,\psi_m)$ converges, and equals $\Haus{\operatorname{dom}(\bm f)}$ if the sum diverges.

The conjecture is presented as an improvement of a proposition on divergence for algebraic points near manifolds; the paper does not provide a resolution.