Bogomolov–Karzhemanov–Kuyumzhiyan conjecture on birational stable flexibility

Let $X$ be an algebraic variety. It is birationally stably flexible if the field extension $\mathbb K(X)(y_1,\ldots,y_n)$ admits a flexible model.

Bogomolov–Karzhemanov–Kuyumzhiyan conjecture. Every unirational algebraic variety $X$ is birationally stably flexible.

This conjecture would strengthen the known implication from stable rationality to birational stable flexibility. The source gives no resolution, so the conjecture remains open.