Maximal extension conjecture for the paraboloid

Let $S$ be the relevant two-dimensional paraboloid and let $E_Sf$ denote its extension operator. For $p,q\in[1,\infty]$, use the mixed norm $L^q_{x_1}L^\infty_{x_2}$. Maximal extension conjecture. If $q\in(3,4]$ and $2/q+1/p<1$, then

$$\|E_Sf\|_{L^q_{x_1}L^\infty_{x_2}}\lesssim\|f\|_p.$$

This is the maximal-extension analogue of the refined maximal Schrödinger conjecture. The paper notes that the estimate holds when $p=4$ and proves it when $q>18/5$; the full range remains open.