Determine the asymptotic limsup of brick chain filtration counts

Determine the value of the limit superior of the normalized supremum of logarithmic brick chain filtration counts, namely \(\limsup_{m\to\infty} F(m)/m^2\), where \(F(m)\) is the supremum of \(\log {}_A(M)\) over finite-dimensional algebras \(A\) and \(A\)-modules \(M\) of composition length \(m\).

Background

The paper defines $F(m)$ as the supremum of the logarithm of the number of brick chain filtrations over all finite-dimensional algebras and modules of composition length $m$. Combining the universal upper bound with the three-arrow Kronecker-algebra construction, it establishes the inequalities $$(1-\log 2)/25\leq \limsup_{m\to\infty}F(m)/m^2\leq \log 2$$.

The unresolved problem is to determine the exact asymptotic constant lying between these two bounds. The question concerns the sharp quadratic-order growth rate of the maximal number of brick chain filtrations as composition length tends to infinity.