Existence conjecture for connected higher Auslander Nakayama algebras

Let $n\geq 2$ and let $k$ satisfy $2\leq k\leq 2n-2$. A connected Nakayama algebra is a connected finite-dimensional Nakayama algebra, and $\operatorname{gldim}(A)$ denotes its global dimension. Existence conjecture. For every such $n$ and $k$, there exists a connected Nakayama algebra $A$ with $n$ simple modules such that $A$ is a higher Auslander algebra and

$$\operatorname{gldim}(A)=k.$$

Computations in the paper verify the corresponding union of spectra for $2\leq n\leq 14$, but the assertion for every $n\geq 2$ remains open in the supplied text.