Do we know a number $n\gt 5$ with no twin prime $n\lt q\lt 2n$? This is essentially a Bertrand's postulate version for twin primes. I am interested in both an explicit example and large lower bounds for it because of this answer of mine. In the comments below the answer, it is shown that there is no such $n$ below $8\times 10^{15}$.
An efficient algorithm would be as follows: take an initial point $m$ for which Bertrand's postulate for twin primes is true (say, $13$). Find the greatest prime twin $p\lt 2n$. The new initial point is $p$. Iterate.
An explicit example of such $n$ would cause a very large gap $\approx n$. Although it seems quite unlikely for such $n$ to exist, a proof remains far from reality, so I am interested in a computational effort.

Do we know $n\gt 5$ with no twin prime $n\lt q \lt 2n$?  If not, what's the best known lower bound?


We can't yet prove that there are infinitely many twin primes, so we certainly can't prove that there's a twin prime in (n, 2n). But it's surely true. Indeed, between n and 2n we expect there to be about $Cn/\log^2 n$ twin primes for some positive constant $C$.
If you look at the worst case, consider A113275: Kourbatov found that the twin prime 1121784847637957 was followed by a longer gap of non-twin primes than any smaller twin prime, but the distance to the next twin prime was just 23382. That's a lot smaller than the 1121784847637957 you'd need to make the twin prime Bertrand fail!
Using the same approach as Michael Stocker I was able to check that there are no exceptions up to $10^{120}.$
Edit: I extended the range to $10^{262}$ and proved all the primalities. Time required was a few hours.
