Conjecture on Betti cones of hypersurface rings

Let $Q$ be a hypersurface ring of embedding dimension $n$ and multiplicity $d$. Write $(\beta_i)_{i\geq 0}$ for a formal rational Betti sequence, with standard basis vectors $\epsilon_i$ in the ambient sequence space.

Betti-cone conjecture. The cone $\mathrm{B}_{\mathbb{Q}}(Q)$ is an $(n+1)$-dimensional cone, and its closure is defined by the following $(n+2)$ extremal rays:

$$\epsilon_0,$$ $$\epsilon_i+\epsilon_{i+1}\quad (i=0,\dots,n-2),$$ $$\frac{d-1}{d}\epsilon_{n-2}+\sum_{i=n-1}^\infty\epsilon_i, \qquad \frac{1}{d}\epsilon_{n-2}+\sum_{i=n-1}^\infty\epsilon_i.$$

This would give a description depending only on the embedding dimension and multiplicity of the hypersurface ring, extending the preceding description of the total hypersurface cone; the conjectural description for a fixed hypersurface ring remains to be established.