Observe that for any $\epsilon > 0$ there are infinitely many triples of $c^\epsilon$-smooth coprime positive integers $a$, $b$ and $c$ such that $a + b = c$. -- Considering triples of the form $(2^n-1,1,2^n)$ and the factorizations of the polynomials $x^n-1 \in \mathbb{Z}[x]$ into cyclotomic polynomials, this holds since the set of quotients $n/\varphi(n)$ for positive integers $n$ is unbounded.

How much can this obvious observation be improved, i.e. how much can the smoothness bound $c^\epsilon$ be lowered such that there are still infinitely many such triples which satisfy that bound?

Or to be more concrete: is there an $\epsilon > 0$ such that there are infinitely many triples of $e^{(\ln c)^{1-\epsilon}}$-smooth coprime positive integers $a$, $b$ and $c$ satisfying $a + b = c$? -- And if yes, which is the supremum of the set of values of $\epsilon$ for which this holds?