Hausdorff measure of hairs without endpoints in the exponential family

Devaney and Krych showed that for $0<\lambda<1/e$ the Julia set of $\lambda e^z$ consists of pairwise disjoint curves, called hairs, which connect finite points, called the endpoints of the hairs, with $\infty$. McMullen showed that the Julia set has Hausdorff dimension $2$ and Karpi\'nska showed that the set of hairs without endpoints has Hausdorff dimension $1$. We study for which gauge functions the Hausdorff measure of the set of hairs without endpoints is finite.