Bérczi's polynomial quotient conjecture for Thom-polynomial coefficients

Let the Thom generating function be

$$\mathrm{Tp}_k(z_1,\ldots,z_k)=\frac{\prod_{m<l}(z_m-z_l)\,Q_k(z_1\ldots z_k)}{\prod_{m+r\le l\le k}(z_m+z_r-z_l)}.$$

For coefficient multi-indices $i_1,\ldots,i_k$, consider neighbouring coefficients obtained by increasing $i_l$ by one and decreasing $i_m$ by one.

Bérczi's neighbouring-coefficient conjecture. The quotient

$$\frac{\operatorname{coeff}_{z_1^{i_1}\ldots z_k^{i_k}}\mathrm{Tp}_k}{\operatorname{coeff}_{z_1^{i_1}\cdots z_l^{i_l+1}\cdots z_m^{i_m-1}\cdots z_k^{i_k}}\mathrm{Tp}_k}<k^2.$$

The conjecture is motivated by background and experimental evidence cited by the source. No proof or disproof is given there.