One-kill-coarsening codimension conjecture for upper triangular Lie algebras

Let $\eta_1$ and $\eta_2$ be gradings on $UT_n^{(-)}$. Say that $\eta_1$ and $\eta_2$ correspond to a 1-kill-coarsening when passing from one grading to the other moves exactly one nontrivial homogeneous basis element into the trivial component. Write $c_m(UT_n^{(-)},\eta)$ for the graded codimension in degree $m$. One-kill-coarsening codimension conjecture. If $\eta_1$ and $\eta_2$ correspond to a 1-kill-coarsening, then, for every $m$,

$$c_m(UT_n^{(-)},\eta_2)=c_m(UT_n^{(-)},\eta_1)+1.$$

The conjecture is motivated by the stated examples for $UT_2^{(-)}$ and certain elementary gradings on $UT_3^{(-)}$, but the supplied text gives no resolution in general.