Keeps changing his mind

Henry decides one morning to do a workout and walks $\frac{3}{4}$ of the way from his home to his gym. The gym $2$ kilometers away from Henry's home. At that point, he changes his mind and walks $\frac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\frac{3}{4}$of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|$ ?

A. $\frac{2}{3}$

B.​ $1$

C. $\frac{6}{5}$

D. $\frac{5}{4}$

E. $\frac{3}{2}$