Integral rank-one conjecture for K-theoretic BPS spaces

For $n\in\mathbb{N}$ and the pairs $(nd,nv)$ under consideration, let $\mathrm{P}(nd)_{nv}$ denote the K-theoretic BPS space of primitive elements, defined by

$$\mathrm{P}(nd)_{nv}:=\operatorname{ker}\left(\bigoplus_{\substack{a+b=n\\\a,b\geqslant 1}}\Delta_{a,b}:K_T(\mathbb{T}(nd)_{nv})\to\bigoplus_{\substack{a+b=n\\\a,b\geqslant 1}}K_T(\mathbb{T}(ad)_{av}\otimes\mathbb{T}(bd)_{bv})\right).$$

Integral rank-one conjecture. The $\mathbb{K}$-module $\mathrm{P}(nd)_{nv}$ is free of rank one.

This is the integral version of the established statement that the localized space $\mathrm{P}(nd)_{nv,\mathbb{F}}$ is one dimensional. The source notes that the torsion-free version for $(d,v)=(1,0)$ and $n=2$ follows from the discussion cited there; the integral assertion for all relevant pairs remains conjectural.