Higher-dimensional projection dimension conjecture

Let $E\subset\mathbb{R}^d$ and let $\mathcal{V}\subset G(d,d-1)$ be Borel sets such that $\dim_{\mathcal{H}}\mathcal{V}>0$ and $\mathcal{V}^{\perp}$ is not contained in a great circle. Higher-dimensional projection dimension conjecture. There exists $V\in\mathcal{V}$ such that

$$\dim_{\mathcal{H}}\pi_V(E)\geq \min\left\{\frac{(d-1)\dim_{\mathcal{H}}E+\dim_{\mathcal{H}}\mathcal{V}}{d},\dim_{\mathcal{H}}E,d-1\right\}.$$

This conjecture proposes a sharp lower bound for the dimension of an orthogonal projection onto at least one hyperplane in a positive-dimensional, non-planar family of directions. The preceding construction shows that the corresponding upper-bound exponent is attained for Cartesian products, while the general lower bound remains open.