The lower-bound conjecture for generalized Tesler matrices

For integers $n$ and $k$ with $n\geq k\geq 11$, let $T(1^k,0^{n-k})$ denote the number of upper-triangular generalized Tesler matrices whose hook-sum vector is $(1^k,0^{n-k})$.

Generalized Tesler-matrix lower-bound conjecture. One has

$$T(1^k,0^{n-k})\geq (k+1)^{n-1}.$$

This conjecture strengthens the preceding eventual lower-bound motivation for generalized Tesler matrices. The supplied text does not state whether it has been resolved.