Spectrum conjecture for connected Nakayama algebras

Let $n\geq 2$ and let $k$ satisfy $2\leq k\leq 2n-2$. A Nakayama algebra is a finite-dimensional algebra whose indecomposable modules are uniserial; it is connected when its underlying algebra is connected, and it is linear or cyclic according to whether its ordinary quiver is a linearly oriented or oriented cyclic quiver. A higher Auslander algebra is an algebra whose dominant dimension equals its global dimension. Spectrum conjecture. For every $n\geq 2$ and $k$ with $2\leq k\leq 2n-2$, there exists a connected, linear or cyclic Nakayama algebra with $n$ simple modules that is a higher Auslander algebra with global dimension $k$. This conjecture concerns the spectrum of possible global dimensions for higher Auslander Nakayama algebras with a fixed number of simple modules; the source notes that it was verified computationally for $n\leq 14$, while the general assertion remains open.