Nuclearity of induced morphisms between colimit Banach rings

For $0<r_2<r_1<1$, let

$$B_r=[?$$

be the Banach ring constructed as the non-expanding colimit of the rings $\mathbb{Z}\{(x/r)^{1/n}\}$ under the morphisms $\alpha_{n,m}$. Nuclearity conjecture. The induced morphism

$$\underset{n}{\operatorname{colim}}^{\leq 1}\mathbb{Z}\{(x/r_1)^{1/n}\}\longrightarrow \underset{n}{\operatorname{colim}}^{\leq 1}\mathbb{Z}\{(x/r_2)^{1/n}\}$$

is nuclear for all $r_2<r_1<1$. The claim concerns the nuclearity of transition morphisms used in the construction of the Banach-algebraic version of the Fargues–Fontaine curve; the supplied text does not indicate whether it has been proved or remains open.