Can every even integer greater than four be written as a sum of two twin primes?

In fact, it was already a conjecture; mathworld says, "It is conjectured that every even number is a sum of a pair of twin primes except a finite number of exceptions whose first few terms are $2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518,\cdots$" ... (OEIS A007534; Wells 1986, p. 132).

There are infinitely many even integers greater than four, so your conjecture would imply that there are infinitely many twin primes. Considering that the twin prime conjecture still has not been solved, I highly doubt that you will be able to prove your conjecture.