Constructing prime $\mathbb{Q}$-Fano threefolds of codimension four via key varieties related with ${\mathbb{P}}^2\times {\mathbb{P}}^2$-fibrations

In our previous research, we constructed the affine varieties ${\mathrm{\Sigma }}_{\mathbb{A}}^{13}$ and ${\mathrm{\Pi }}_{\mathbb{A}}^{14}$ whose partial projectivizations admit ${\mathbb{P}}^2\times {\mathbb{P}}^2$-fibrations with relative Picard number one. In this paper, we produce prime quasi-smooth $\mathbb{Q}$-Fano 3-folds which are anticanonically embedded of codimension four and belong to 23 (resp.8) classes in the Graded Ring Database [GRDB], as weighted complete intersections in weighted projectivizations of ${\mathrm{\Sigma }}_{\mathbb{A}}^{13}$ (resp.${\mathrm{\Pi }}_{\mathbb{A}}^{14}$ or its cone). We also show that a general member of the anticanonical linear system of a general prime $\mathbb{Q}$-Fano $3$-fold constructed in this way is a quasi-smooth $K3$ surface with at worst Du Val singularities.