The exceptional-set projection conjecture in the plane

Let $A ewcommand{\mathbb{R}}{\mathbb{R}}\subset \mathbb{R}^2$ have $\dim(A)=a$. For $\theta ewcommand{\Pi}{\Pi}\in G(1,\mathbb{R}^2)$, let $\Pi_\theta:\mathbb{R}^2\rightarrow\theta$ be the orthogonal projection onto the line $\theta$. For $0<s<\max\{1,a\}$, define

$$E_s(A):=\{\theta:\dim(\Pi_\theta(A))<s\}.$$

The exceptional-set projection conjecture. Then

$$\dim(E_s(A))\leq\max\{0,2s-a\}.$$

This is the Euclidean analogue of the paper's finite-field exceptional-set estimates and predicts a sharp bound for directions in which the projection of $A$ has dimension less than $s$. The source does not state a resolution, so the conjecture is recorded as open.