Comparison of local oscillation quantities for weighted BMOA spaces

Let $v$ be an admissible weight, let $X$ be either $\operatorname{BMOA}_v$ or $\operatorname{VMOA}_v$, and assume $X\not\subset H^\infty$. Let $\delta_z$ denote evaluation at $z$, let $\gamma(\psi,a,r)$ be the local quantity used in the paper, and let $g$ and $\phi$ be the associated data. The local-oscillation comparison conjecture. One has

$$\sup_{a\in\mathbb D}\|\delta_{\phi(a)}\|_{X^*}\gamma(\psi,a,2)\lesssim_{v,g,\psi,\phi}\sup_{a\in\mathbb D}\|\delta_{\phi(a)}\|_{X^*}\gamma(\psi,a,1)$$

and

$$\limsup_{|\phi(a)|\to1}\|\delta_{\phi(a)}\|_{X^*}\gamma(\psi,a,2)\lesssim_{v,g,\psi,\phi}\limsup_{|\phi(a)|\to1}\|\delta_{\phi(a)}\|_{X^*}\gamma(\psi,a,1).$$

Such comparisons would relate the two scales of local oscillation appearing in the compactness criteria for weighted composition operators.