A noncircular oval with convex unit-tangent iterates

For an oriented regular plane curve $γ$, define ${\cal T}γ=γ+τ$, where $τ$ is its unit tangent. We construct a noncircular curve $Γ_0$ such that every iterate ${\cal T}^nΓ_0$, $n\geq0$, is a smooth embedded closed curve with everywhere positive curvature. This answers a question of Tabachnikov. The