Dimension lower-bound conjecture for sums of projected planar sets and sets on a line

Let $K_{1}\subset\mathbb{R}^{2}$ and $K_{2}\subset\mathbb{R}$ be analytic sets, with $\wp_{\theta}(K_{1})$ denoting the projection of $K_{1}$ in direction determined by $\theta$. Assume

$$\dim_{\mathrm{H}}K_{1}+\dim_{\mathrm{H}}K_{2}\leq 1.$$

Projected-sum dimension conjecture. For almost every $\theta\in(0,2\pi)$,

$$\dim_{\mathrm{H}}\bigl(\wp_{\theta}(K_{1})+K_{2}\bigr)\geq \dim_{\mathrm{H}}K_{1}+\dim_{\mathrm{H}}K_{2}.$$

This is presented as a weaker variant of the paper's dimension-conservation conjecture. The preceding proposition proves a related lower bound for product sets and the special family of projections, but the stated projected-sum claim remains open in the supplied text.