Let $M^3$ be a closed, connected and oriented smooth $3$-manifold, and fix an integer $g \geq 1$. Is it true that for a generic set of Riemannian metrics on $M$ the set of closed, connected and orientable embedded minimal surfaces of genus $g$ in $M$ is compact? If not, is it true for a generic set of metrics of positive scalar curvature?

There is no hope that this holds for every metric, since $\Sigma_{\gamma} \times \mathbb{S}^1$ has a sequence of minimal embedded tori winding around $\mathbb{S}^1$ as many times as one wishes. Here $\Sigma_\gamma$ is a compact Riemannian surface of genus $\gamma \geq 1$ endowed with a metric of constant curvature equal to $0$, if $g=1$, or equal to $-1$ if $g \geq 2$ (see this paper, for instance).