Is Zaremba's conjecture true?

For finite continued fractions in which all partial quotients lie in the alphabet $\{1,2,3,5\}$, it is shown that the set of denominators not exceeding $N$ has cardinality $\gg N^{0.85}$. A calculation using an analogue of Bourgain-Kontorovich's theorem from 2011 gives $\gg N^{0.80}$.
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