Let $f \in L^1 \cap L^\infty(\mathbb R)$ and $K \in BV(\mathbb R)$. Do these assumptions suffice to prove that for the convolution $K \ast f$ we have that $$K \ast f \in W^{1,\infty}(\mathbb R)$$ holds?

From this post, I know that this convolution is BV, but is it true that it is also $W^{1,\infty}$?