Full-dimensional projection conjecture for non-degenerate families in three dimensions

Let $J\subset\mathbb{R}$ be an open interval, and let $\gamma:J\to S^{2}$ be a $\mathcal{C}^{3}$ curve satisfying

$$\operatorname{span}\{\gamma(\theta),\dot{\gamma}(\theta),\ddot{\gamma}(\theta)\}=\mathbb{R}^{3},\qquad \theta\in(0,1).$$

Set $\ell_{\theta}:=\operatorname{span}(\gamma(\theta))$ and $V_{\theta}:=\ell_{\theta}^{\perp}$, and write $\rho_{\theta}:=\pi_{\ell_{\theta}}$ and $\pi_{\theta}:=\pi_{V_{\theta}}$ for the corresponding orthogonal projections. Let $B\subset\mathbb{R}^{3}$ be an analytic set.

Full-dimensional projection conjecture. For almost every $\theta\in J$,

$$\dim\rho_{\theta}(B)=\min\{\dim B,1\}\quad\text{and}\quad \dim\pi_{\theta}(B)=\min\{\dim B,2\}.$$

The conjecture strengthens the known lower bounds for non-degenerate one-dimensional families of line and plane projections. It predicts the maximal dimension allowed by the target spaces, but the source provides no evidence of a resolution.