Bounded-height conjecture for nondegenerate points in fiberwise small group subschemes

Let $\mathcal{A}\rightarrow S$ be the abelian scheme in the paper, and let $\mathcal{X}\subseteq\mathcal{A}$ be a subvariety. For $t\in\mathbb{N}$, let $\mathcal{A}_{(\leq t)}$ be the union of all group subschemes in the fibers of dimension at most $t$, and let $\mathcal{X}^{\mathrm{deg}}(t)$ denote the corresponding degenerate locus. Bounded-height conjecture. The closed points of

$$\left(\mathcal{X}\setminus\mathcal{X}^{\mathrm{deg}}(t)\right)\cap\mathcal{A}_{(\leq t)}$$

form a set of bounded total height. This is proposed as an idealistic extension of the paper's main bounded-height theorem, allowing arbitrary rather than only flat group subschemes; the supplied context gives no resolution.