Fässler–Orponen projection conjecture for non-degenerate curves on the sphere

Let $\gamma:[0,1]\to\mathbb S^2$ be a $C^2$ curve satisfying

$$\det(\gamma,\gamma',\gamma”)(\theta)\ne 0,\quad \forall\,\theta\in[0,1].$$

For each $\theta$, let $p_{\gamma(\theta)}$ denote projection onto the line in direction $\gamma(\theta)$, and let $\pi_{\gamma(\theta)}$ denote projection onto the plane orthogonal to $\gamma(\theta)$. Fässler–Orponen's conjecture. For every Borel set $A\subseteq\mathbb R^3$,

$$\dim p_{\gamma(\theta)}(A)=\min\{1,\dim A\},\qquad \dim \pi_{\gamma(\theta)}(A)=\min\{2,\dim A\}$$

both hold for $\mathcal L^1$-almost every $\theta\in[0,1]$. This is a restricted projection conjecture for directions lying on a non-degenerate curve on the sphere; the supplied text attributes it to Fässler and Orponen but gives no resolution, so its status is left open.