Let $A$ be a representation-finite algebra and $M$ an indecomposable module with finite projective dimension $g >0$.
Question 1: Do we have $dim(Ext_A^g(M, \tau_g(M)))=1$? Here $\tau_g(M)=\tau ( \Omega^{g-1}(M))$.
Question 2: Could this even be true when $A$ is not represenation-finite? Probably not, but the computer did not find a counterexample yet.
Question 1: I think that the answer is no. We have $$\operatorname{Ext}^g_A(M,\tau_g(M)) \simeq \operatorname{Ext}^1_A(\Omega^{g-1}_A(M), \tau(\Omega^{g-1}_A(M)).$$
Look at the following example in QPA:
gap> Q := DynkinQuiver( "A", 5, [ "l", "l", "r", "r" ] );
<quiver with 5 vertices and 4 arrows>
gap> KQ := PathAlgebra( GF(17), Q );
<GF(17)[<quiver with 5 vertices and 4 arrows>]>
gap> arrows := ArrowsOfQuiver( Q ) * One( KQ );
[ (Z(17)^0)*a_1, (Z(17)^0)*a_2, (Z(17)^0)*a_3, (Z(17)^0)*a_4 ]
gap> a1 := arrows[ 1 ];
(Z(17)^0)*a_1
gap> a2 := arrows[ 2 ];
(Z(17)^0)*a_2
gap> a3 := arrows[ 3 ];
(Z(17)^0)*a_3
gap> a4 := arrows[ 4 ];
(Z(17)^0)*a_4
gap> rels := [ a2 * a1, a3 * a4 ];
[ (Z(17)^0)*a_2*a_1, (Z(17)^0)*a_3*a_4 ]
gap> A := KQ / rels;
<GF(17)[<quiver with 5 vertices and 4 arrows>]/<two-sided ideal in <GF(17)[<quiver with 5 vertices and 4 arrows>]>,
(2 generators)>>
gap> S3 := SimpleModules( A )[ 3 ];
<[ 0, 0, 1, 0, 0 ]>
gap> g := ProjDimensionOfModule( S3, 4 );
2
gap> OmegaS3 := NthSyzygy( S3, g - 1 );
<[ 0, 1, 0, 1, 0 ]>
gap> DTrOmegaS3 := DTr( OmegaS3 );
<[ 1, 0, 0, 0, 1 ]>
gap> ext := ExtOverAlgebra( OmegaS3, DTrOmegaS3 );
[ <<[ 1, 0, 0, 0, 1 ]> ---> <[ 1, 1, 0, 1, 1 ]>>,
[ <<[ 1, 0, 0, 0, 1 ]> ---> <[ 1, 0, 0, 0, 1 ]>>, <<[ 1, 0, 0, 0, 1 ]> ---> <[ 1, 0, 0, 0, 1 ]>> ], function( map ) ... end ]
gap> Length( ext[ 2 ] );
2
