The conjecture that the line-intersection bound is tight

Let $d\ge 2$ and $n\ge 2$. Let $f_d(n)$ denote the largest volume of a $d$-dimensional open set with $n$ connected components and no pair of points at an integral distance. Let $l_d(n)$ denote the maximum volume of a $d$-dimensional open set with $n$ connected components, each of diameter at most $1$, such that its intersection with every line has total length at most $1$. Also let $S_d$ be the unit $d$-dimensional ball and let $\lambda_d$ denote $d$-dimensional volume. The line-intersection conjecture. For all $n\ge 2$ and $d\ge 2$,

$$l_d(n)=n\cdot\lambda_d(S_d).$$

This conjecture proposes that the known upper bound for $f_d(2)$ extends to the relaxed quantity $l_d(n)$; the paper establishes the corresponding values of $f_d(n)$ but leaves the claimed equality for $l_d(n)$ open.