Optimality of the inhomogeneous rational-approximation constant

Let $a,b,r,s$ be integers as in the $(a,b,r,s)$-approximation setting, and let $\theta$ denote the constant on the right-hand side of the relevant approximation inequality. The condition is that $r\neq 0$ or $s\neq 0$. Optimality conjecture. If $r\neq 0$ or $s\neq 0$, the constant $ab/4$ appearing on the right-hand side of the inequality cannot be improved uniformly in $\xi\in\mathbb{R}\backslash\mathbb{Q}$. This concerns the sharp uniform constant in inhomogeneous rational approximation; the source notes that the assertion is known in the non-trivial parity case $a=b=2$ with $r\neq0$ or $s\neq0$, while the general case is left open.