Projection formulation of the arithmetic Kakeya conjecture

For $A\subset\mathbb{Z}\times\mathbb{Z}$ and $r\in\mathbb{Q}\cup\\{\infty\\}$, define

$$\pi_r(A)=\\{x+ry:(x,y)\in A\\},$$

with $x+\infty y=y$. Projection formulation of the arithmetic Kakeya conjecture. For every $\epsilon>0$, there exist $r_1,\ldots,r_k\in\mathbb{Q}\cup\\{\infty\\}$, none equal to $-1$, such that for every $A\subset\mathbb{Z}\times\mathbb{Z}$,

$$|\pi_{-1}(A)|\leq\max_{i=1,\ldots,k}|\pi_{r_i}(A)|^{1+\epsilon}.$$

The source presents this as another equivalent formulation of the arithmetic Kakeya conjecture. Its status is not resolved in the supplied text.