Rudolph's characterization conjecture for quasipositive ribbons

Let $D^4(r)=\{(z,w)\in\mathbb{C}^2:(|z|^2+|w|^2)^{1/2}\leq r\}$. A smoothly embedded surface $S\subset D^4(r)$ is ribbon-embedded if $\partial S=S\cap\partial D^4(r)$ and the restriction of $(|z|^2+|w|^2)$ to $S$ is a Morse function with no local maxima on $\operatorname{Int}S$. If $r'$ is a regular value with $0<r'<r$, set $S'=S\cap D^4(r')$; a pair $(S,S')$ obtained this way is a ribbon-embedded pair. A surface ambient isotopic to a ribbon-embedded surface is a ribbon, and a subsurface $S'\subset S$ of a ribbon is a subribbon if $(S,S')$ is ambient isotopic to a ribbon-embedded pair. For sufficiently small $\varepsilon\neq0$, let $\widetilde T_{n,n}$ be any one of the mutually isotopic ribbon-embedded surfaces $\Gamma_n(\varepsilon)\cap D^4(1)$, where

$$\Gamma_n(\varepsilon)=\{(z,w)\in\mathbb{C}^2:z^n+w^n=\varepsilon\}.$$

Rudolph's characterization conjecture. A ribbon is quasipositive if and only if, for some $n$, it is ambient isotopic to a subribbon of $\widetilde T_{n,n}$. This would characterize quasipositive ribbons through the standard complex plane-curve surfaces $\widetilde T_{n,n}$; the source presents it as a conjectural generalization, and no resolution is supplied here.