The bounded leftover conjecture for ordered graph tilings

Let $H$ be an ordered graph. An $H$-tiling in an ordered graph $G$ is a collection of vertex-disjoint copies of $H$ in $G$, and let $\delta(G)$ denote the minimum degree of $G$. Write $\chi^*_{cr}(H)$ for the modified critical chromatic number of $H$. Bounded leftover conjecture. There is a constant $C=C(H)\in\mathbb N$ such that, for every $n$-vertex ordered graph $G$ satisfying

$$\delta(G)\geq\left(1-\frac{1}{\chi^*_{cr}(H)}\right)n,$$

$G$ contains an $H$-tiling covering all but at most $C$ vertices.

This is an ordered-graph analogue of a theorem of Shokoufandeh and Zhao and would strengthen the asymptotic threshold result by showing that only a bounded number of vertices need remain uncovered under the critical minimum-degree condition. The supplied text does not state whether the conjecture has been resolved.