Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category. In proposition 1.2. of https://link.springer.com/article/10.1007%2FBF01195702 the following was proven:

Let A be a piecewise hereditary algebra with $n$ simple modules and an indecomposable module $X$. Then $pdim(X)+idim(X) \leq n+1$.

It was noted there that the authors do not know whether the bound is attained.

Question: Is it now known whether the bound is attained?

I do not even know any acyclic algebra where the bound is attained.

Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category. In proposition 1.2. of https://link.springer.com/article/10.1007%2FBF01195702 the following was proven:

Let A be a piecewise hereditary algebra with $n$ simple modules and an indecomposable module $X$. Then $pdim(X)+idim(X) \leq n+1$.

It was noted there that the authors do not know whether the bound is attained.

Question: Is it now known whether the bound is attained? In case it is not known, is there an example where it is attained or is the optimal bound $n$ (attained for example at the Kroenecker algebra)?

I do not even know any acyclic algebra where the bound is attained.

Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category. In proposition 1.2. of https://link.springer.com/article/10.1007%2FBF01195702 the following was proven:

Let A be a piecewise hereditary algebra with $n$ simple modules and an indecomposable module $X$. Then $pdim(X)+idim(X) \leq n+1$.

It was noted there that the authors do not know whether the bound is attained.

Question: Is it now known whether the bound is attained?

I do not even know any acyclic algebra where the bound is attained.

Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category. In proposition 1.2. of https://link.springer.com/article/10.1007%2FBF01195702 the following was proven:

Let A be a piecewise hereditary algebra with $n$ simple modules and an indecomposable module $X$. Then $pdim(X)+idim(X) \leq n+1$.

It was noted there that the authors do not know whether the bound is attained.

Question: Is it now known whether the bound is attained?

I do not even know any acyclic algebra where the bound is attained.