Dimension conservation conjecture for non-degenerate projections

Let $B\subset\mathbb{R}^{3}$ be an analytic set, and let $(\rho_{\theta})_{\theta\in U}$ and $(\pi_{\theta})_{\theta\in U}$ be non-degenerate families of projections onto lines and planes, respectively. Proposition 1 states that

$$\dim_{\mathrm{H}} B\leq \tfrac12\implies \dim_{\mathrm{H}}\rho_{\theta}(B)=\dim_{\mathrm{H}}B$$

and

$$\dim_{\mathrm{H}} B\leq 1\implies \dim_{\mathrm{H}}\pi_{\theta}(B)=\dim_{\mathrm{H}}B$$

almost surely. Dimension conservation conjecture. In Proposition 1(a), the hypothesis $\dim_{\mathrm{H}} B\leq \tfrac12$ can be relaxed to $\dim_{\mathrm{H}} B\leq 1$. In part (b), the hypothesis $\dim_{\mathrm{H}} B\leq 1$ can be relaxed to $\dim_{\mathrm{H}} B\leq 2$. This predicts dimension preservation for almost every projection throughout the full natural ranges suggested by the target dimensions, beyond the currently established thresholds.