The maximum-filling formula conjecture for mixed even and odd board dimensions

Let $\bar d(r,c)$ denote the maximum number of non-bonding dominoes that can be placed on an $r\times c$ rectangular board. Maximum-filling conjecture. If one of $r$ and $c$ is an odd multiple of $2$ and the other is odd, then

$$\bar d(r,c)=\frac{rc}{4}+\frac{1}{2}.$$

This is part of a proposed parity-based description of the maximum filling, inferred from the computed data and figures; its validity beyond the displayed cases remains open.