Hi TourRook,
I used to supply 2000*1000, 3072*1562 but for FS its much better to use 4096 * 2048 (binary multiplier used by Immervision).
Try to shareb a little when downsized to the final output resolution and add a little saturation (5-15%) & contrast (4-8%) before sharpening.
It helped me a lot.
Of course it depends on the quality of the input image- but as a general rule it might do.