Order-wise sample-complexity conjecture for monotone test functions

Let $f(\cdot)$ be a monotone test function, let $h(f)$ denote the corresponding function appearing in the sample-complexity bounds, and let $\Gamma(q)$ denote the corresponding quantity optimized over $q\in(0,1)$. In the sparse regime with $d=n^\theta$ and $0\leq\theta<1$, the sample-complexity bounds are expressed in terms of these quantities.

Sample-complexity conjecture. For any monotone test function $f(\cdot)$,

$$\frac{\min_{q\in(0,1)}\Gamma(q)}{h(f)d\log n}\in\Theta(1).$$

A positive resolution would tightly characterize the order-wise sample complexity for every monotone test function, extending the order-wise tightness already established for noisy test functions. The supplied text does not indicate whether this conjecture has been resolved.