Logarithmic lower-bound conjecture for monotonicity testing in the EVAL model

Let $n$ be the size of the ordered domain and let $\varepsilon>0$ be the distance parameter. In the ${\sf EVAL}$ model, a tester accesses a distribution through evaluation queries.

Logarithmic lower-bound conjecture. Monotonicity testing in the ${\sf EVAL}$ model has query complexity

$$\Omega\left(\frac{\log n}{\varepsilon}\right).$$

The conjecture asserts that the logarithmic dependence on $n$ in the known upper bound is tight. The context gives lower bounds combining to $\Omega\left(\max\left(\log n/\log\log n,1/\varepsilon\right)\right)$, but does not establish the conjectured logarithmic dependence.