Bernoulli-moment sign conjectures for isolated hypersurface singularities

Let $f:({\mathbb C}^{n+1},0)\to({\mathbb C},0)$ be an isolated hypersurface singularity with spectral-number moment series

$$V^{\operatorname{sing}}(f)=\sum_{k=0}^{\infty}V_{2k}^{\operatorname{sing}}(f)\frac{t^{2k}}{(2k)!}.$$

For a formal series $V=\sum_{k=0}^{\infty}V_{2k}t^{2k}/(2k)!$, define the Bernoulli moments by

$$\Gamma^{\operatorname{Ber}}(V,\nu)=\sum_{k=0}^{\infty}\Gamma_{2k}^{\operatorname{Ber}}(V,\nu)\frac{t^{2k}}{(2k)!}=V\cdot\exp\left(\nu\log\frac{t/2}{\sinh(t/2)}\right).$$

Bernoulli-moment sign conjectures. For every $k\in{\mathbb N}$, both the weak form

$$(-1)^k\Gamma_{2k}^{\operatorname{Ber}}(V^{\operatorname{sing}}(f),n+1)>0$$

and the strong form

$$(-1)^k\Gamma_{2k}^{\operatorname{Ber}}(V^{\operatorname{sing}}(f),\alpha_\mu-\alpha_1)\geq0$$

should hold. The weak form is explicitly weaker than the strong form in the source. These conjectures are central to the paper and generalize the variance inequality to all Bernoulli moments; no resolution is supplied.