  The description of the $\eta$ and $\eta^\prime$ mesons in the Dyson-Schwinger
approach has relied on the Witten-Veneziano relation. The present paper
explores the consequences of using instead its generalization recently proposed
by Shore. On the examples of three different model interactions, we find that
irrespective of the concrete model dynamics, our Dyson-Schwinger approach is
phenomenologically more successful in conjunction with the standard
Witten-Veneziano relation than with the proposed generalization valid in all
orders in the $1/N_c$ expansion.
