The equality criterion for the left-hand Hausdorff-dimension bound

Let $k\geq 1$ be an integer and let $\mathscr{C}$ be a curve as in the paper. Write $\mathscr{H}^{k}_{\lambda}$ for the set of points satisfying the relevant Diophantine approximation condition, and let $\Pi_{1}$ denote projection onto the first coordinate. The left-hand inequality in the dimension bound referred to in the source is attained under a condition on $\lambda$. Equality conjecture. The condition $\lambda\geq 1$ is necessary and sufficient for equality in the left-hand inequality in the dimension bound. This conjecture concerns the unresolved parameter ranges discussed immediately before it, in particular the cases $k=2$ with $\lambda\in(1,t)$ and $k\geq 3$ with $\lambda\in(1/k,t]$; the source explains that the expected analogue may fail for smaller $\lambda$.