The versal-family pullback question for p-divisible groups

Let $X$ be a complete curve over $\mathbb{F}_{q}$, and let $\mathscr{G}\to X$ be a $p$-divisible group with ordinary and supersingular points. A $p$-divisible group is everywhere versally deformed when its deformation is versal at every point.

Versal-family pullback conjecture. Does there exist a variety $Y$ and a height-$2$, dimension-$1$ $p$-divisible group $\mathscr{H}$ on $Y$, everywhere versally deformed, together with a map $X\to Y$ such that $\mathscr{G}$ is the pullback of $\mathscr{H}$?

\xymatrix{\mathscr{G}\ar[d]\ar[r] & \mathscr{H}\ar[d]\\\\ X\ar[r] & Y }

The question asks whether such groups can be pulled back from a universal versally deformed family. The source notes that $Y$ cannot in general be required to be a curve, citing examples of fake Hilbert modular surfaces.