An approach to the finitistic dimension conjecture

Let $R$ be a finite dimensional $k$-algebra over an algebraically closed field $k$ and $\mathrm{m}\mathrm{o}\mathrm{d}R$ be the category of all finitely generated left $R$-modules. For a given full subcategory $\mathcal{X}$ of $\mathrm{m}\mathrm{o}\mathrm{d}R,$ we denote by $\text{pfd}\mathcal{X}$ the projective finitistic dimension of $\mathcal{X}\mathrm{.}$ That is, $\pfd $\mathcal{X}$:=$\mathrm{s}\mathrm{u}\mathrm{p}$ {\pd X : X\in$\mathcal{X}$ $\text{and}$ \pd X<\infty}.$
It was conjectured by H. Bass in the 60’s that the projective finitistic dimension $\text{pfd}(R):=\text{pfd}(\mathrm{m}\mathrm{o}\mathrm{d}R)$ has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and J. Todorov defined a function $\mathrm{\Psi }:\mathrm{m}\mathrm{o}\mathrm{d}R\to \mathbb{N},$ which turned out to be useful to prove that $\text{pfd}(R)$ is finite for some classes of algebras. In order to have a different approach to the finitistic dimension conjecture, we propose to consider a class of full subcategories of $\mathrm{m}\mathrm{o}\mathrm{d}R$ instead of a class of algebras, namely to take the class of categories $\text{F}(\theta )$ of $\theta$-filtered $R$-modules for all stratifying systems $(\theta ,\le )$ in $\mathrm{m}\mathrm{o}\mathrm{d}R\mathrm{.}$