Let \(N_q(g)\) denote the maximum number of \(\mathbb{F}_q\)-rational points on a smooth, absolutely irreducible curve of genus \(g\) defined over \(\mathbb{F}_q\). Of basic importance in the asymptotic theory of such curves (or function fields) is \[ A(q)=\limsup_{g\rightarrow\infty} {N_q(g)\over g}. \] M. Perret [J. Number Theory 38, 300-322 (1991; Zbl 0741.11044)] described a method to obtain better lower bounds for \(A(q)\) than a bound due to Serre, but his method depended on a conjecture concerning whether certain towers of function fields were in fact infinite towers. The authors show that Perret’s conjecture fails by giving a counterexample based on the theory of narrow ray class fields.