Oberlin's conjecture on Hausdorff dimensions of unions of affine lines

Let $A(d,1)$ denote the space of affine lines in $cmathbb{R}^d$. Suppose $1\leq k\leq d-1$ is an integer and let $\beta\in[0,1]$. If $\mathcal{A}\subset A(d,1)$ satisfies

$$\dim_{\mathcal{H}}\mathcal{A}\geq 2(k-1)+\beta,$$

Oberlin's conjecture. Then

$$\dim_{\mathcal{H}}\bigcup_{l\in\mathcal{A}}l\geq k+\beta.$$

This conjecture concerns lower bounds for the Hausdorff dimension of unions of affine lines and is attributed in the source to D. Oberlin. The stated bound is sharp for general families of affine planes in the related result described in the introduction; the conjecture itself is presented without a resolution in the source.