Continuous bounded variation path that fails to be Lipschitz

Question

Let $C^{1\text{-var}}([0,T];\mathbb{R}^d)$ denote the space of continuous bounded variation paths taking values in $\mathbb{R}^d$. Similarly, let $C^{1\text{-Höl}}([0,T];\mathbb{R}^d)$ denote the space of $1$-Hölder continuous paths. Now, to me it's clear that $$C^{1\text{-Höl}}([0,T];\mathbb{R}^d)\subset C^{1\text{-var}}([0,T];\mathbb{R}^d).$$ I believe the opposite inclusion fails. Do you agree? If so, can you help me arrive at an example? Any insights will be appreciated.