Is the vanishing Ext_R^n(Hom_R(L,N),R)=0 automatic under the symmetry setup?

Ascertain whether the following implication holds without additional assumptions: for a Cohen–Macaulay local ring R and finitely generated modules M and N with PR(M,R) < ∞, PR(N,R) < ∞, b_M < ∞, and L an MCM syzygy of M, if PR(M,N) < ∞ then Ext_R^n(Hom_R(L,N), R) = 0 for all sufficiently large n.

Background

Theorem 5.4(1) proves a module-specific symmetry of Ext vanishing direction under several hypotheses, including the explicit vanishing condition Ext_R^n(Hom_R(L,N), R) = 0 for n ≫ 0, to conclude that PR(N,M) < ∞ from PR(M,N) < ∞.

The question asks whether this vanishing prerequisite is actually a consequence of the other stated assumptions (finite Auslander bounds and finite PR with R), potentially simplifying the criteria needed for the Ext symmetry implication.