conjectures - 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.