Iyama's finiteness theorem via strongly quasi-hereditary algebras

Let $\mathrm{\Lambda }$ be an artin algebra and $X$ a finitely generated $\mathrm{\Lambda }$-module. Iyama has shown that there exists a module $Y$ such that the endomorphism ring $\mathrm{\Gamma }$ of $X\oplus Y$ is quasi-hereditary, with a heredity chain of length $n$, and that the global dimension of $\mathrm{\Gamma }$ is bounded by this $n$. In general, one only knows that a quasi-hereditary algebra with a heredity chain of length $n$ must have global dimension at most $2n-2$. We want to show that Iyama’s better bound is related to the fact that the ring $\mathrm{\Gamma }$ he constructs is not only quasi-hereditary, but even left strongly quasi-hereditary: By definition, the left strongly quasi-hereditary algebras are the quasi-hereditary algebras with all standard left modules of projective dimension at most~1.