Inequality for weighted homogeneous isolated singularities

Prove that, for every positive integer e, every weight vector w=(w_1,...,w_n) with positive integer entries, and every w-weighted homogeneous polynomial g of degree d_0 defining a hypersurface with an isolated singularity, the inequality \[\sum_{i: e\nmid w_i,\ e\nmid d_0-w_i}\left(\frac{w_i}{e}-\left\lfloor\frac{w_i}{e}\right\rfloor\right)\geq\sum_{i: e\nmid w_i,\ e\nmid d_0-w_i}\left(\frac{d_0}{e}+\left\lfloor1-\frac{d_0}{e}\right\rfloor\right)\frac{w_i}{d_0}\] holds.

Background

The paper introduces this conjecture after proving the exponential-sum estimate under the sufficient condition d_0≥κ(w). The authors explain that removing this degree condition requires a deeper understanding of the relationship between the weighted degree, the weight vector, and the singular loci of coordinate restrictions of the polynomial.

The conjectured inequality is designed to replace the numerical estimates that use d_0≥κ(w). The subsequent proposition states that, if this conjecture holds, then Igusa's conjecture for exponential sums follows for the corresponding weighted homogeneous polynomial.