Linear equivalence conjecture for hereditary and rooted quasirandomness

Let $H$ be a graph, let $p\in[0,1]$, and let $G$ be a graph. The property $\mathcal{P}_{H,p}^{*}(\delta)$ means that every $U\subseteq V(G)$ satisfies

$$c(H,G[U])=p^{e(H)}|U|^{v(H)}\pm\delta v(G)^{v(H)}.$$

Let $\mathcal{R}_{H,p}(\varepsilon)$ denote the rooted quasirandomness condition defined in the paper, requiring the appropriate labelled-copy estimates for vertex subsets associated with the vertices of $H$. Linear equivalence conjecture. For every graph $H$, every $p\in[0,1]$, and every $\delta>0$,

$$\mathcal{P}_{H,p}^{*}(\delta)\implies\mathcal{R}_{H,p}(\varepsilon)$$

for some $\varepsilon=O_{H,p}(\delta)$. The paper states that this conjecture remains open and that, together with its main theorem, it would suffice to establish linear equivalence between $\mathcal{P}_{H,p}^{*}(\varepsilon)$ and $\mathcal{R}_{H,p}(\varepsilon)$.