A coefficient-ratio bound for Thom-polynomial coefficients

Let $n$ be a positive integer. Let $\mathbf{i}$ be a multiindex with at least one negative component and with $\sum_i i_i=n^2$. For multiindices $\mathbf{a},\mathbf{b}\geq\mathbf{0}$ satisfying

$$\sum_i a_i=\sum_i b_i,\qquad \mathbf{a}\cdot\mathbf{b}=0,$$

write $\mathrm{Tp}_{\mathbf{z}^{\mathbf{u}}}$ for the coefficient of the Laurent monomial $\mathbf{z}^{\mathbf{u}}$ in the Thom generating function.

Thom-polynomial coefficient-ratio conjecture. One has

$$\frac{\mathrm{Tp}_{\mathbf{z}^{-\mathbf{i}}}}{\mathrm{Tp}_{\mathbf{z}^{-\mathbf{i}+\mathbf{a}-\mathbf{b}}}}<n^{\sum_i b_i}.$$

The supplied text introduces this bound as a statement needed for an estimate and gives no resolution status beyond that context.