Is the algebraic Grothendieck group of a weighted projective space finitely generated ?

In our paper with N. Pavic we have proved that over an algebraically closed field of char. 0, if the weights $a_0, \dots, a_n$ are coprime, so that singularities of $\mathbb{P}(a_0, \dots, a_n)$ are isolated, then $K_0(\mathbb{P}(a_0, \dots, a_n)) = \mathbb{Z}^{n+1}$, see Application 3.2 in

https://arxiv.org/pdf/1809.10919.pdf

Our proofs rely on comparison of $K_0(X) = K_0(Perf(X))$ with $G_0(X) = K_0(D^b(X))$ using $K$-groups of Orlov's singularity category and completion arguments. The same methods apply to compute $K_0(X)$ for any quasi-projective variety $X$ with isolated quotient singularities.

I never did determine whether this was true or not (it was this point that unfortunately required me to use the BOT construction in my paper).

However, at the time I was working on this, it was my suspicion that the torsion part of the $KH_{0}$ of a WPS would be 0. If true, then (at least in characteristic 0) my paper would imply that the question of whether or not $K_{0}$ of a WPS is finitely generated boils down to whether or not its $(\mathcal{F}_{K})_{0}$ is finitely generated.

However, if $KH_{0}$ has torsion, then the problem could potentially take on a whole new level of complexity (or could be equivalent, depending on whether or not it can be shown that the torsion part of $KH_{0}$ is at least controlled).

I hope someone does come along to pick this up again. I no longer work in math professionally, but I would be very interested in seeing further progress on this. -Adam