Independence of the elliptic parameter in higher-rank K-theoretic DT theory

Let $r$ and $k$ be integers, and let $obreak\boldsymbol{T}_k\subset\boldsymbol{T}_1$ be the subtorus defined by $\mathfrak t^{\frac12}=e^{\pi i k/r}$. Write

$$\mathsf{DT}_{r,k}^{\rm ell}(\mathbb{A}^3,q,t,w;p)=\left.\mathsf{DT}_{r}^{\rm ell}(\mathbb{A}^3,q,t,w;p)\right|_{\boldsymbol{T}_k}.$$

Independence conjecture. The series $\mathsf{DT}_{r,k}^{\rm ell}(\mathbb{A}^3,q,t,w;p)$ does not depend on the elliptic parameter $p$.

This conjecture extends the parameter-independence established on the subtori with $k\in r\mathbb{Z}$, where the series is a power of the MacMahon function. The general case remains open.