Moduli conjecture for conic bundles associated with $F^{2+}_5$

Let $C$ be the canonically embedded complete-intersection curve of degree $8$ and genus $5$ arising as the base locus of a net of quadrics in $\mathbb{P}^4$, let $\pi(C)\subset\mathbb{P}^3$ be its projection from a chosen point of $C$, and set

$$X=\operatorname{Bl}_{\pi(C)}\mathbb{P}^3.$$

Consider conic bundles over $\mathbb{P}^2$ birational to $X$ over $\mathbb{P}^2$ and with fixed anticanonical degree. Moduli conjecture. Their moduli space is the curve $C$. This is the corresponding geometric conjecture for conic bundles in type $F^{2+}_5$; the source gives no resolution status.