Surjectivity of the genus function for weight-2 newforms on $\Gamma_0(N)$

Let $g_0^+(N,2)$ denote the dimension of the space of weight-$2$ newforms on $\Gamma_0(N)$, equivalently the genus of the corresponding new modular curve. Surjectivity conjecture. For every nonnegative integer $k$, there is at least one positive integer $N$ such that

$$g_0^+(N,2)=k.$$

Computations for $N\le132{,}000$ found every nonnegative integer through $4{,}361$ among the values of $g_0^+(N,2)$, motivating this assertion; its validity for all $k$ remains open.