Fässler–Orponen Marstrand-type projection conjecture for curved families in three dimensions

Let $J\subset\mathbb{R}$ be a bounded open interval, let $\gamma\colon J\to S^2$ be a $\mathcal{C}^2$ curve satisfying

$$\operatorname{span}\{\gamma(\theta),\dot{\gamma}(\theta),\ddot{\gamma}(\theta)\}=\mathbb{R}^3,\qquad \theta\in J,$$

and let $\rho_\theta\colon\mathbb{R}^3\to\mathbb{R}$ be the orthogonal projection onto the line spanned by $\gamma(\theta)$. Fässler–Orponen's Marstrand-type projection conjecture. If $K\subset\mathbb{R}^3$ is a Borel set, then

$$\dim_{\mathrm{H}}\rho_\theta(K)=\min\{\dim_{\mathrm{H}}K,1\}$$

for almost every $\theta\in J$. The conjecture is a curved-family analogue of Marstrand's projection theorem and was posed as the first part of Fässler and Orponen's Conjecture 1.6. The present paper proves this assertion, so it is solved.