Extrapolated multilevel variance-decay conjecture

Let $\prescript{\mathrm{Ext}}{}{\Phi}^{\ell}$ and $\prescript{\mathrm{Ext}}{}{\Psi}^{\ell}$ denote the extrapolated fine- and coarse-level quantities at level $\ell$. Assume that $c_1,c_2\in\mathbb{R}$. Extrapolated multilevel variance-decay conjecture. The per-level variance satisfies

$$\operatorname{Var}\left[\prescript{\mathrm{Ext}}{}{\Phi}^{\ell}-\prescript{\mathrm{Ext}}{}{\Psi}^{\ell}\right]\lesssim 2^{-2\ell}\max\left(c_1 2^{-2\ell},c_2 2^{-2\ell}\right)=\mathcal{O}(2^{-4\ell}).$$

The extrapolated antithetic construction is intended to raise the time-difference contribution to second order, yielding the stated fourth-order variance decay, while leaving the computational cost per sample unchanged.