If $\gamma(b)$ is not conjugate to $\gamma(a)$ along $\gamma$. Show that $I(V, V)=I(J, J)$ if and only if $V=J .$
There are a number of problems with your argument. As @Didier commented, it does not follow from the absence of conjugate points that $\gamma$ is length-minimizing. More importantly, most of your claims about $I(V,V)$ and $I(J,J)$ would be justified if $V$ and $J$ were proper (i.e., vanishing at the endpoints), but there's no such assumption in the problem statement.
BTW, in case anyone wants to look this up in the book, this is Problem 10-12, not Exercise 10.12. The problems are at the ends of the chapters, while exercises are incorporated into the text of the chapters.