The first conjecture implies the second.
Whether the second conjecture is true was first asked in [KT]. Both conjectures were stated in [BJW].
The second conjecture is implied by Jørgensen’s conjecture, which asserts that every -connected -minor-free graph is apex (which have minimum degree at most and are thus not -connected). Since Jørgensen’s conjecture is true for sufficiently large graphs [KNTWa,KNTWb], the second conjecture is true for sufficiently large graphs.
Bibliography
*[BJW] János Barát, Gwenaël Joret, David R. Wood. Disproof of the List Hadwiger Conjecture, Electronic J. Combinatorics 18:P232, 2011.
*[KT] Ken-ichi Kawarabayashi and Bjarne Toft. Any 7-chromatic graph has or as a minor. Combinatorica 25 (3), 327–353, 2005.
[KNTWa] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. minors in -connected graphs of bounded tree-width. http://arxiv.org/abs/1203.2171
[KNTWb] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. minors in large -connected graphs. http://arxiv.org/abs/1203.2192
* indicates original appearance(s) of problem.