Exceptional sets for length under restricted families of projections onto lines in ${\mathbb{R}}^3$

It is shown that if $A\subseteq {\mathbb{R}}^3$ is a Borel set of Hausdorff dimension $\dim A>1$, and if ${\rho }_{\theta }$ is orthogonal projection to the line spanned by $(\cos \theta ,\sin \theta ,1)$, then ${\rho }_{\theta }(A)$ has positive length for all $\theta$ outside a set of Hausdorff dimension at most $\frac{3-\dim A}{2}$.