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.
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?