Exceptional-set conjecture for orthogonal projections

Fix $0\leq t/2\leq s\leq t\leq 1$, and let $K\subset\mathbb{R}^2$ be a Borel set with $\dim_{\mathrm{H}}K\geq t$. For $e\in S^1$, let $\pi_e:\mathbb{R}^2\to\mathbb{R}$ be the orthogonal projection onto the line spanned by $e$. Exceptional-set conjecture for orthogonal projections.

$$\dim_{\mathrm{H}}\\{e\in S^1:\dim_{\mathrm{H}}\pi_e(K)<s\\}\leq 2s-t.$$

This would improve Kaufman's bound under the stronger assumption $\dim_{\mathrm{H}}K\geq t>s$; the sharpness of Kaufman's bound in that regime is stated to be open.