Asymptotic-density-zero conjecture for dimensions of weight-2 newspaces

For $N\geq 1$, let $S_2^{\textnormal{new}}(N)$ denote the newspace of cuspidal modular forms of weight $2$ and level $N$, and set

$$A=\{\dim S_2^{\textnormal{new}}(N)\}_{N\geq 1}.$$

Asymptotic-density-zero conjecture. The set $A$ has asymptotic density $0$ in the natural numbers. The corresponding density statement is known for the dimensions $\{\dim S_2(N)\}_{N\geq 1}$, but the supplied text says that the newspace case is not known.