The $K_{6}$ forbidden-minor conjecture for first Symanzik reducibility

Let $K_{6}$ be the complete graph on six vertices. A graph is a forbidden minor for reducibility with respect to the first Symanzik polynomial if it is not reducible with respect to that polynomial, while each proper minor is reducible.

$K_{6}$ forbidden-minor conjecture. The complete graph $K_{6}$ is a forbidden minor for reducibility with respect to the first Symanzik polynomial.

This conjecture proposes a specific obstruction to first-Symanzik reducibility; the supplied source gives no resolution or further evidence for it.