Where's the list of connections of conjectures list on Wikipedia?
There's an OEIS sequence that partially connects Collatz and Goldbach, as 2 mod 4 numbers having Goldbach partitions formed via it's members, should imply a Goldbach partition for the number around the bend.
Twin prime implies, infinitely many 0 mod 12 numbers have Goldbach partitions,
Green-Tao ( a Theorem not a Conjecture) implies infinitely many arithmetic progressions of even numbers have Goldbach partitions.
Cunningham chains of first kind, imply that if $n$ has a Goldbach partition where both aren't end members ( or the only ones in lentth 1 chains ...) then 2n+2 will have one ...
As you can see, There are quite a few even towards Goldbach.