# On a conjecture about tilting modules

There is the following conjecture on tilting modules (see also History of an open problem on partial tilting modules):

Given a partial tilting module $$T$$ over a finite dimensional algebra $$A$$ (that is $$Ext_A^i(T,T)=0$$ for all $$i \geq 1$$ and $$pd(T) < \infty$$), then $$T$$ is a tilting module iff $$|T|=n$$ where $$n$$ denotes the number of simple $$A$$-modules and $$|T|$$ the number of non-isomorphic indecomposable summands of $$T$$.

The conjecture is true for tilting modules of projective dimension one and thus for all hereditary algebras.

Questions:

Is this conjecture known for algebras of global dimension 2 (or any finite global dimension)?

Is the conjecture known for tilting modules of projective dimension 2?

Is this conjecture known for representation finite algebras?