Let ${\rm rad}(n)$ denote the radical of a positive
integer $n$, i.e. the product of its distinct prime divisors.
Given positive integers $a$ and $b$, the triple $(a,b,a+b)$ is
called an *abc triple* if $a$ and $b$ are coprime and
${\rm rad}(ab(a+b)) < a+b$. The *quality* of an abc triple
$(a,b,a+b)$ is defined as the quantity
$$
q(a,b,a+b) \ := \ \frac{\log{(a+b)}}{\log{({\rm rad}(ab(a+b)))}},
$$
and the abc conjecture asserts that for any $\epsilon > 0$
there are only finitely many abc triples of quality greater than $1+\epsilon$.

Now let the *smoothness* of an abc triple $(a,b,a+b)$ be
$$
s(a,b,a+b) \ := \ \frac{\log{(\log{(\frac{a+b}{{\rm rad}(ab(a+b))})})}}{\log{(\log{(a+b)})}}.
$$
Clearly the smoothness of any abc triple is strictly smaller than $1$.

Question:Is it true that for any $\epsilon > 0$ there are abc triples with smoothness $\geq 1-\epsilon$?

*Examples:*

The triple $(1, 8, 9)$ has quality $1.22629$ and smoothness $-1.14676$.

The triple $(3, 125, 128)$ has quality $1.42657$ and smoothness $0.23562$.

The triple $(10, 2187, 2197)$ has quality $1.28975$ and smoothness $0.26825$.

The triple $(1, 2400, 2401)$ has quality $1.45567$ and smoothness $0.43400$.

The triple $(1, 4374, 4375)$ has quality $1.56789$ and smoothness $0.52238$.

The triple $(343, 59049, 59392)$ has quality $1.54708$ and smoothness $0.56635$.

The triple $(3200, 4823609, 4826809)$ has quality $1.46192$ and smoothness $0.57855$.

The triple $(2, 6436341, 6436343)$ has quality $1.62991$ and smoothness $0.65457$.

The triple $(283, 8251953125, 8251953408)$ has quality $1.58076$ and smoothness $0.67991$.

The triple $(125, 11174240024064, 11174240024189)$ has quality $1.53671$ and smoothness $0.690851$.

The triple $(24833, 5020969537415167, 5020969537440000)$ has quality $1.62349$ and smoothness $0.73326$.

The triple $(8654525279998779296875, 229727166528260169448321941, 229735821053540168227618816)$

has quality $1.48053$ and smoothness $0.72594$.