Birational factorization conjecture for elliptic fibrations on weighted Fano threefold hypersurfaces

Let $X$ be the weighted Fano threefold hypersurface considered in the paper, with numerical parameter $N$, and let $a_{2}$ and $a_{3}$ be the weights defining the weighted projective plane $f4a0(1,a_{2},a_{3})$. Let

$$\rho:X\dashrightarrow\mathbb{P}^{2}$$

be a rational map such that the normalization of a general fiber of $\rho$ is an elliptic curve. Birational factorization conjecture. If $N\notin\{3,60,75,84,87,93\}\cup\Omega$, where $\Omega=\{1,2,7,9,11,17,19,20,26,30,36,44,49,51,64\}$, then there is a commutative diagram

\xymatrix{ &&&&X\ar@{-->}[dll]_{\psi}\ar@{-->}[dr]^{\rho}&&&\\ &&\mathbb{P}(1,a_{2},a_{3})\ar@{-->}[rrr]_{\phi}&&&\mathbb{P}^{2},&&}

where $\psi$ is the natural projection and $\phi$ is a birational map. This asserts that the elliptic fibration factors birationally through the natural projection to $\mathbb{P}(1,a_{2},a_{3})$.