Foul or fair

[clevbrown]

I would not normally ask this question because I feel I have it correct and the rule book backs me, but I am being questioned by an experienced high school baseball umpire and I want to make sure I have it right. So after reading the rule books, I am going to the "Experts" on this web site to confirm what I think or let you all convince me I am wrong.

This is NFHS Baseball but I believe it is the same answer for all levels of softball and baseball.

Is this statement true or false: A batted ball that lands without being touched in the batter's box is always a foul ball.

I believe this is false. There is a space in the batters box on the pitcher side and the home plate side (front, inside) that is in fair territory. I am being told the entire batter box is considered in foul teritory, but I have looked at the NFHS baseball, ASA SOftball, and Babe Ruth Baseball rule books and I find nothing to support this claim.

NFHS states:
A foul is a batted ball
a. which settels on foul territory between home and first base or between home and third base or
b. that bounds past first or third base on or over foul territory or
c. that first falls on foul territory beyond first or third base or
d. that while on or over foul territory, touches the person of an umpire or a player or an object foreign to the natural ground or
e. touches the ground after inadvertantly being declared foul by an umpire.

Am I right or not?