  Let $E \subseteq \mathbb{R}^n$ be a union of line segments and $F \subseteq
\mathbb{R}^n$ the set obtained from $E$ by extending each line segment in $E$
to a full line. Keleti's \textit{line segment extension conjecture} posits that
the Hausdorff dimension of $F$ should equal that of $E$. Working in
$\mathbb{R}^2$, we use effective methods to prove a strong packing dimension
variant of this conjecture, from which the generalized Kakeya conjecture for
packing dimension immediately follows. This is followed by several doubling
estimates in higher dimensions and connections to related problems.
