Conjectural strengthened Hausdorff-dimension bounds for best-approximation sets

Let $\Theta_n(w)$ denote the set used in the paper for the parameter $w$, and let $\Theta^n$ be the corresponding union over $w\in[n,\infty]$. For $n\ge2$ and $w\in[n,\infty]$, the proposed strengthened bounds are

$$\dim_H(\Theta_n(w))\ge n-2+\frac{2}{w^2+1},\qquad \dim_H(\Theta^n)>n-2+\frac{2}{n^2+1}.$$

Conjectural refinement. These inequalities should hold for all $n\ge2$ and $w\in[n,\infty]$. This would improve the established lower bounds, whose corresponding numerators are $1$ rather than $2$, and the source presents the refinement as unsettled.