There is a good reason to check! I had seen a source that said 10 to 12 GPM/cuft and I figured you would want the high end of that.
http://www.home-water-purifiers-and-filters.com/calgon-activated-carbon.php is not the source I used initially, but it does name a specific media. Incidentally, you probably do want coconut shell activated carbon. Another term to research.
Found a better source: See the graph on the upper right of
http://www.calgoncarbon.com/wp-content/uploads/2015/08/DS-OLC12x4015-EIN-E1.pdf I see people say you normally want 30% to 40% bed expansion during backwash. Note the red line for 72F. That says about 9 gpm/cuft... (splitting the distance between 30% and 40%. The 5600SXT could do that with the 12 inch tank. The tanks are sized with a 50% "freeboard" which leaves room for bed expansion during backwash.
If you had even warmer water normally, imagine a new line that is even lower than the red line. There is probably a formula or graph somewhere for water that is even warmer. But your water may not be warmer on average, so it looks like that 5600SXT could do it based on the Calgon graph. 7 GPM through a 12 inch tank is 8.912 gpm/ft2, so that seems to get you in range.
You know the issues. Let us know what additional stuff you learn.