The convex-combination version of Freiman's theorem for planar blockers

Let $P$ be a set of points in the plane with at most $\frac{1}{2}|P|$ points collinear. Suppose that a set $B$ blocks $P$, meaning that for every two distinct points $x,y\in P$ there is a real number $\gamma\in(0,1)$ such that $\gamma x+(1-\gamma)y\in B$. A $d$-dimensional progression is a set of the form $\{v_0+x_1v_1+\cdots+x_dv_d:x_i\in[1,n_i]\}$ for suitable vectors and integer ranges. General Freiman conjecture. Then $P$ is contained in a $d$-dimensional progression of size at most $\beta|P|$, for some $d$ and $\beta$ depending only on $\alpha$, where $|B|\leq\alpha|P|$. The conjecture is presented as a convex-combination analogue of Freiman's theorem that would imply the superlinear blocking conjecture.