The non-degenerate smooth Roth conjecture for thick Cantor sets

Let $K\subset\mathbb{R}$ be a Cantor set satisfying $\tau(K)>1$, and let $f$ be a continuously differentiable function satisfying $f(0)=0$ and $f'(0)>0$. Non-degenerate smooth Roth conjecture. There exist $x\in\mathbb{R}$ and $t>0$ such that

$$\{x-t,x,x+f(t)\}\subset K.$$

The conjecture would remove the additional derivative bounds required by the paper's perturbative proof of its non-linear Roth theorem, extending the result to every continuously differentiable function with positive derivative at the origin. Its status is not resolved in the supplied source.