Does any sum of twin primes, where the sum is greater than 12, also represents the sum of 2 other distinct primes?

I was in the midst of proving a conjecture when I came across an observation that led me to forming a potentially new conjecture. The conjecture goes as follows:

Any given sum of twin primes (specifically the two primes in a twin prime pair, ie. $11$ and $13$) where the sum is greater than $12$, also represents the sum of 2 other distinct primes.

I've proven this for the first $1000$ twin primes and my computer is calculating beyond that set. Anyways, does anybody have any ideas of how I could go about proving this conjecture? I apologize if this conjecture has already been posed.

• This OEIS sequence lists how many ways even integers can be written as sums of primes - and it suggests every even integer greater than $12$ is the sum of at least two distinct pairs of primes, which would imply your conjecture. Of course, we have no idea how to prove Goldbach's conjecture, and I would assume that your conjecture is not much easier. – Milo Brandt Feb 15 '15 at 1:17
• Interesting, I noticed by hand calculation that it looked as if it applied to other numbers but I was hesitant to pose such as topic without examining it further. Thanks for the link! – user1939991 Feb 15 '15 at 1:20
• It's easy to come up with conjectures about prime numbers. But aye, proving them, that's the tough part. – Mr. Brooks Feb 15 '15 at 1:45
• See my blog ideasfornumbertheory.wordpress.com for some possible insights. – Sylvain Julien Feb 16 '15 at 21:08
• Also a special case of this mathSE conjecture. – 6005 Jul 18 '16 at 19:20

There is a conjecture which states that the total number of writing $n$ as a sum of two odd primes is $\sim \frac{n}{2(\ln n)^2}$.