The differentiable support-point length conjecture for planar curves

Let $\gamma:[a,b]\rightarrow\mathbb{E}^2$ be an arbitrary continuous rectifiable parametric curve such that, for every $t\in(a,b)$, there is a non-zero derivative vector $\gamma'(t)$. Let $L:[a,b]\rightarrow\mathbb{R}$ denote the length of the arc of $\gamma$ corresponding to the parameter interval $[a,t]$, and let $T(t)$ be the set of support-point parameters associated with $t$. Differentiable support-point length conjecture. The inequality

$$\varlimsup_{t\to a}\frac{\sup\{L(\tau)\mid \tau\in T(t)\}}{L(t)}\geq\frac{1}{e}$$

holds. This is presented as a version of the preceding open question for curves with a non-zero derivative at every interior parameter value; its resolution is not given in the source.