Negative covariance conjecture for scrambled $(0,m,s)$-nets

Let $\tilde P_n$ be a scrambled $(0,m,s)$-net in base $b$, and let $f\in L^2([0,1)^s)$ be a function whose base $b$ Walsh series decomposition satisfies

$$\sigma_{\mathbf{k}}^2(f)=a^r x^k\alpha_f,$$

where $a\in[0,1]$, $x\in[0,1)$, $k=k_1+\cdots+k_s$, $\alpha_f$ is a positive constant depending on $f$, and $r$ is the number of nonzero coordinates of $\mathbf{k}\in\mathbb{N}^s$. Negative covariance conjecture.

$$\operatorname{Cov}(f(\mathbf{U}_I),f(\mathbf{U}_J))\leq 0$$

and

$$\operatorname{Var}(\hat I_n(f))\leq\operatorname{Var}(\hat I_{MC,n}(f)).$$

This conjecture generalizes the result proved for $a=(b-1)/b$. It would establish nonpositive covariance between distinct scrambled-net points and variance no greater than that of the corresponding Monte Carlo estimator, but the authors report that their proof technique does not extend to other values of $a$ and that only experimental evidence supports the claim.