Hasse principle conjecture for hypersurfaces in products of projective spaces

Let $d_i,n_i$ be positive integers for $i\in\{1,\ldots,k\}$, and consider multidegree $(d_1,\ldots,d_k)$ hypersurfaces in $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_k}$ over $\mathbb{Q}$. A hypersurface is everywhere locally soluble if it has a point over every completion of $\mathbb{Q}$, and is soluble if it has a rational point. Hasse principle conjecture. If $d_i>n_i+1$ for every $i$, then $100\%$ of everywhere locally soluble hypersurfaces fail the Hasse principle. If $d_i<n_i+1$ for some $i$, except when $(k,n_1,n_2,d_1,d_2)=(2,1,2,\geq2,2)$ or $(2,2,1,2,\geq2)$, then $100\%$ of everywhere locally soluble hypersurfaces satisfy the Hasse principle, and the proportion of soluble hypersurfaces equals $c=\prod_\nu c_\nu$. The conjecture predicts the typical global behaviour of locally soluble hypersurfaces in products of projective spaces; the stated exceptional cases remain outside its conclusion.