A continuity-based criterion for failure of property (iv)

Let $A\subset\mathbb{R}^2$ and suppose there exists $y\in A$ such that

$$\mathcal{H}^{1}(y\cap B_{\rho}(A))=\infty\quad\text{for all }\rho>0.$$

For some $\rho_1>0$, assume that $u\in C^0(\mathbb{R};\mathbb{R})$, $G_y\in G(1,2)$ is the rotation of $\mathbb{R}^2$ onto $\mathbb{R}$, and

$$y\in G_y^{-1}(\operatorname{graph}u)\cap B_{\rho_1}(y),$$ $$\overline{A\cap G_y^{-1}(\operatorname{graph}u)}=G_y^{-1}(\operatorname{graph}u)\subset A.$$

The continuity-based criterion. Under these assumptions, $A$ does not have property (iv). The conjecture proposes that the conclusion of Theorem 2 remains valid when the relevant graph is merely continuous, even if all the infinite measure is produced by a single function. The source presents this as a possible improvement and gives heuristic geometric reasons, but no proof or resolution is supplied.