Length of sets under restricted families of projections onto lines

Let $\gamma :I\to S^2$ be a $C^2$ curve with $\det (\gamma ,{\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }\mathrm{\prime }})$ nonvanishing, and for each $\theta \in I$ let ${\rho }_{\theta }$ be orthogonal projection onto the span of $\gamma (\theta )$. It is shown that if $A\subseteq {\mathbb{R}}^3$ is a Borel set of Hausdorff dimension strictly greater than 1, then ${\rho }_{\theta }(A)$ has positive length for a.e. $\theta \in I$. This answers a question raised by Käenmäki, Orponen and Venieri.