Here's a Haskell program to display the values for which the conjecture is false.

Within a minute, the largest value it found was the following, rather nice quintuple: $$[468883,468887,468889,468893,468899]$$

```
import Math.NumberTheory.Primes
specialPrime n = ((nthPrime n) + (nthPrime (n + 3)) == (nthPrime (n + 1)) + (nthPrime (n + 2))) && ((nthPrime n) + (nthPrime (n + 4)) == (nthPrime (n + 2)) + (nthPrime (n + 3)))
specialPrimes = filter (specialPrime) [1..]
specialTuples n = let v = specialPrimes !! (n-1)
in fmap nthPrime [v..(v + 4)]
checkConjecture n = isPrime (foldr (+) 0 $ specialTuples n)
falseValues = fmap specialTuples $ filter (not . checkConjecture) [1..]
main = do
sequence_ $ fmap print falseValues
```