Positive measure for restricted projections along smooth non-degenerate curves

Let $1\le m<n$, and let $\gamma :[0,1]\to {\mathbb{R}}^n$ be a smooth non-degenerate curve. For $\theta \in [0,1]$, set $$V_{\theta }:=\mathrm{span}\{{\gamma }^{\mathrm{\prime }}(\theta ),\dots ,{\gamma }^{(m)}(\theta )\},$$ and let ${\pi }_{\theta }$ denote the orthogonal projection onto $V_{\theta }$. We prove that if $E\subset {\mathbb{R}}^n$ is a Borel set with ${\dim }_{\mathrm{H}}E>m$, then $${\mathcal{H}}^m({\pi }_{\theta }(E))>0$$ for almost every $\theta \in [0,1]$.