Weaker optimality conjecture for the weight classes H for Schrödinger operators

Let $V\in RH_d$. The classes $H_{p,c}^{V,m}$ are weight classes associated with the reverse Hölder potential $V$, and $R_V$ and $T_V^*$ are the Riesz transform and heat maximal operator, respectively. For a weight $w$, $\|R_V\|_{L^p(w)}<\infty$ and $\|T_V^*\|_{L^p(w)}<\infty$ denote boundedness on $L^p(w)$.

Weaker optimality conjecture. There exist $c_1,c_2,m_1,m_2>0$ such that

$$H_{p,c_1}^{V,m_1}\subset\left\{w:\|R_V\|_{L^p(w)}<\infty\right\}\subset H_{p,c_2}^{V,m_2}$$

and

$$H_{p,c_1}^{V,m_1}\subset\left\{w:\|T_V^*\|_{L^p(w)}<\infty\right\}\subset H_{p,c_2}^{V,m_2}.$$

This is presented as a weaker form of the preceding conjecture. The source proves comparison inclusions between the $H$ and $S$ classes, but does not establish these boundedness chains in general.