Increasing Goldbach partitions The Goldbach conjecture states that:

every even number that is greater than 2 is the sum of two primes.

We will consider a Goldbach partition of a number n to be a pair of two primes adding to n.  We are concerned with numbers is of increasing Goldbach partition. We measure the size of a number's Goldbach partitions by the size of the smallest prime in all of that number's partitions.  A number is of increasing partition if this size is greater than the size of all smaller even numbers.
Task
Given an even integer n &gt; 2, determine if n is of increasing Goldbach partition, and output two unique values, one if it is and one if it is not.
This is code-golf, so you should aim to minimize the number of bytes in your source code.
OEIS A025018
Jelly, 12 bytes

ÆRðfạṂ
Ç€M⁼W


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How it works

Ç€M⁼W   Main link. Argument: n

Ç€      Map the helper link over [1, ..., n].
  M     Get all indices of the maximum.
    W   Wrap; yield [n].
   ⁼    Test the results to both sides for equality.


ÆRðfạṂ  Helper link. Argument: k

ÆR      Prime range; get all primes in R := [1, ..., k].
  ð     Begin a dyadic chain with arguments R and k.
    ạ   Absolute difference; yield k-p for each p in R.
   f    Filter; keep the q in R such that q = k-p for some p in R.
     Ṃ  Take the minimum.
        This yields 0 if the array is empty.