Cross Diagonal Cover - II

I will continue the previous post. Now the question to be answered is that:

What percentage (or how many) of given $m\times n$ grid can be covered by following Cross Diagonal Cover Algorithm?

I did some calculations, for example:

As per my calculations, I have following conjecture:

The following 3 cases are possible:

1. If $m = n$, then $m$ squares will be covered.
2. If $m>n$ and both $m,n$ are odd, then $m$ squares will be covered.
3. If at least one of $m,n$ is even then $\frac{mn}{2}$ squares will be covered.

As you can see that Case - 1 is quite trivial. I tried to prove other two cases for one week, but failed, so decided to post it as conjecture.