Armstrong's characteristic-polynomial conjecture for the Tesler poset

Let $U_n$ be the set of $n\times n$ upper-triangular matrices with non-negative integer entries. For $A=(a_{i,j})\in U_n$, define its hook sums by

$$h_k:=\left(a_{k,k}+a_{k,k+1}+\cdots+a_{k,n}\right)-\left(a_{1,k}+a_{2,k}+\cdots+a_{k-1,k}\right).$$

Let $\mathcal{T}(1^n)$ be the set of matrices whose hook sums are all $1$, and let $P(1^n)$ be the Tesler poset on this set. Its characteristic polynomial is

$$\chi(P(1^n);q)=\sum_{A\in P(1^n)}\mu(\hat{0},A)q^{\rho(P(1^n))-\rho(A)}.$$

Armstrong's conjecture. The characteristic polynomial satisfies

$$\chi(P(1^n);q)=(q-1)^{\binom{n}{2}}.$$

This conjecture concerns the enumerative structure of the Tesler poset and was attributed to Drew Armstrong. The supplied text does not state whether it has been resolved.