The planar finite-field Furstenberg set lower-bound conjecture

Fix $s\in(0,1]$ and $t\in(0,2]$. Let $E$ be an $(s,t;2,1)$-set over $\mathbb{F}_p$, meaning that $E$ is a union of subsets $Y(V)$ over a family of affine $1$-planes $V$ containing at least a constant multiple of $p^t$ planes, with each $Y(V)$ containing at least a constant multiple of $p^s$ points. Finite-field Furstenberg lower-bound conjecture. For every $\varepsilon>0$,

\\#E\gtrsim_{\varepsilon}p^{\min\left\\{s+t,\frac{3}{2}s+\frac{1}{2}t,s+1\right\\}-\varepsilon}.

This conjectured lower bound is motivated by the Ren–Wang result and concerns the optimal exponent in the planar finite-field Furstenberg set problem. The supplied text does not state whether the conjecture has been proved or disproved.