Dilation conjecture for affine conic bundles

Let $M$ be an affine conic bundle of the form described in the paper, and suppose that the polynomial $p(z_3,\ldots,z_{n+1})$ is weighted homogeneous and invertible. Dilation conjecture. Then $M$ admits a dilation. This conjecture proposes a structural explanation for the nonexistence of exact Lagrangian $K(\pi,1)$ spaces in these affine conic bundles. The excerpt gives no resolution of the conjecture.