On the uniqueness of solutions to the isotropic $L_{p}$ dual Minkowski problem

We prove that the unit sphere is the only smooth, strictly convex solution to the isotropic $L_p$ dual Minkowski problem \begin{align*} h^{p-1} |D h|^{n+1-q}\mathcal{K}=1, \end{align*} provided $(p,q)\in (-n-1,-1]\times [n,n+1)$.