The redundancy of the curve-intersection assumption in the intrinsic Lipschitz lemma

Let $f:{\mathbb{R}}^d\longrightarrow {\mathbb{R}}^m$ be continuous and piecewise Lipschitz with exceptional set $\Theta$. Suppose that $f$ is intrinsically Lipschitz with respect to the intrinsic metric associated with $\Theta$. The additional assumption that for every $x,y\in {\mathbb{R}}^d$ and $\eta>0$ there exists a continuous curve $\gamma$ from $x$ to $y$ with $\ell(\gamma)<\|x-y\|+\eta$ and $\#(\gamma\cap\Theta)<\infty$ is not necessary to conclude that $f$ is Lipschitz with respect to the Euclidean metric, with the same Lipschitz constant. Redundancy conjecture. Item 1.3 in the assumptions of the intrinsic Lipschitz lemma is not necessary to prove the assertion of the lemma. The claim concerns the proof of a regularity result for functions that are intrinsically Lipschitz across a piecewise-Lipschitz exceptional set; whether the stated geometric assumption can be removed is left unresolved in the supplied text.