Rimányi's nonnegativity conjecture for Thom-polynomial coefficients

Let $k$ be a positive integer, and let

$$\mathrm{Tp}_k(z_1,\ldots,z_k)$$

be the Thom generating function. For a multiindex $\mathbf{i}=(i_1,\ldots,i_k)\in{\mathbb Z}^k$ with $i_1+\cdots+i_k=0$, let $\mathrm{Tp}_{\mathbf{i}}$ denote the coefficient of $z_1^{i_1}\cdots z_k^{i_k}$ in its Laurent expansion in the domain $|z_1|\ll\cdots\ll|z_k|$.

Rimányi's conjecture. For every such multiindex, the coefficient is nonnegative:

$$\mathrm{Tp}_{\mathbf{i}}\geq 0.$$

The paper states that this was proved for $k=1,2,3$, while the general case remains open in the supplied text.