= Solution
Let $H(s,z)$ be the <free homotopy> from $\alpha$ to $\beta$ and put $v(s)=H(s,1)$. Because both endpoint loops are based at $x_0$, this is itself a loop based at $x_0$.
Cut the circle parameter at 1 and regard $H$ as a map on a square. Traversing its boundary gives the concatenation $\alpha*v*\bar\beta*\bar v$. The boundary is null-homotopic since the map extends over the square. Thus, in the <fundamental group>,
$$
[\alpha][v][\beta]^{-1}[v]^{-1}=1,
\qquad \boxed{[\beta]=[v]^{-1}[\alpha][v]}.
$$
Hence the corresponding elements are <conjugate group elements>. The moving basepoint path is precisely what distinguishes free <homotopy> from <based homotopy>.
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