Quadratic commutativity conjecture for D8 $qq$-characters

Let $\mathsf{T}_{\four}^{K}(x)$ denote the D8 $qq$-character and let $\mathsf{f}^{K_{1}|K_{2}}_{\four\four}(x_{1}/x_{2})$ be the corresponding one-loop factor. D8 quadratic commutativity conjecture. For arbitrary $K_{1}$ and $K_{2}$, the quadratic relations are

$$\mathsf{f}^{K_{2}|K_{1}}_{\four\four}\left(x_{2}/x_{1}\right)\mathsf{T}_{\four}^{K_{1}}(x_{1})\mathsf{T}^{K_{2}}_{\four}(x_{2})-\mathsf{f}^{K_{1}|K_{2}}_{\four\four}\left(x_{1}/x_{2}\right)\mathsf{T}_{\four}^{K_{2}}(x_{2})\mathsf{T}_{\four}^{K_{1}}(x_{1})=0.$$

Moreover, this commutativity uniquely determines the sign factor $(-1)^{\sigma_{4}(\rho)}$ up to a global $\mathbb{Z}_{2}$ symmetry. The conjecture has been checked computationally through five instantons, but no proof is currently given.