Geometric interpolation conjecture for EXTRA performance guarantees

Let $E_{x}^{\kappa_1,\kappa_2}(\mathcal{K},\theta)$ denote the performance guarantee after $\mathcal{K}$ iterations of EXTRA with $\theta \mathcal{N}$ agents whose local functions belong to $\mathcal{F}_{\mu_1,L}$ and $(1-\theta)\mathcal{N}$ agents whose local functions belong to $\mathcal{F}_{\mu_2,L}$, where $\theta\in[0,1]$. Let $E_{x}^{\kappa_1}(\mathcal{K})$ and $E_{x}^{\kappa_2}(\mathcal{K})$ denote the worst-case errors after $\mathcal{K}$ iterations of EXTRA when all local functions have uniform conditioning, with $\kappa_1=L/\mu_1$ and $\kappa_2=L/\mu_2$. By definition,

$$E_{x}^{\kappa_1}(\mathcal{K})=E_{x}^{\kappa_1,\kappa_2}(\mathcal{K},1),\qquad E_{x}^{\kappa_2}(\mathcal{K})=E_{x}^{\kappa_1,\kappa_2}(\mathcal{K},0).$$

Geometric interpolation conjecture. For all $\mathcal{K}>0$, $\kappa_1,\kappa_2>0$, and $\theta\in[0,1]$, the mixed-conditioning performance satisfies

$$E_{x}^{\kappa_1,\kappa_2}(\mathcal{K},\theta)=\bigl(E_{x}^{\kappa_1}(\mathcal{K})\bigr)^{\theta}\bigl(E_{x}^{\kappa_2}(\mathcal{K})\bigr)^{1-\theta}.$$

The conjecture proposes that the performance with two equivalence classes of agents is the geometric interpolation of the uniform-conditioning worst-case errors. The surrounding experiment suggests that the guarantee depends on the relative proportions of the two classes rather than on the total number of agents, but the source provides no resolution of this conjecture.