Pillay's conjecture on almost quantifier elimination and largeness

Let $K$ be a field. Say that $K$ has almost quantifier elimination if every formula $\phi(x)$, with $x=(x_1,\ldots,x_m)$, is equivalent to a formula $\exists y\,\theta(x,y)$, where $y=(y_1,\ldots,y_n)$, $\theta$ is quantifier-free, possibly with parameters from $K$, and for some $k$ one has

$$K^{\mathrm{alg}}\models\forall x\,\exists^{\leq k}y\,\theta(x,y).$$

Pillay's conjecture. If $K$ has almost quantifier elimination, then $K$ is large. Many familiar model-complete fields have almost quantifier elimination, including pseudofinite fields and fields that are algebraically, real, or $p$-adically closed; whether the stated implication holds in general is open.