The one-dimensional Sobolev gradient bound for infimal convolution

Let $L$ satisfy the assumptions given in the introduction, let $u$ be the Sobolev function under consideration on $\mathbb{R}$, and let $u_t$ denote its infimal convolution with the scaled Lagrangian. Write $u'$ and $u'_t$ for their one-dimensional derivatives, and let $\|\cdot\|_p$ denote the $L^p$ norm. One-dimensional gradient-bound conjecture. If $n=1$, then

$$\|u'_t\|_p\leq C\|u'\|_p$$

for any $1\leq p\leq\infty$. The conjecture concerns the one-dimensional case left open by the preceding construction, which establishes failure of a uniform bound in dimensions $n\geq 3$; the status of the analogous case $n=2$ is not asserted here.