Algebraic quadruple conjecture for basepoint-free pencils of binary quartics

Let $f=(f_1,f_2)$ be a basepoint-free pencil of binary quartics, viewed as a morphism $f\colon\mathbb{P}^1\to\mathbb{P}^1$. Let $(E,V,\phi,\xi)$ denote an algebraic quadruple, and let its associated morphism be the morphism constructed from this quadruple. Algebraic quadruple conjecture. There exists an algebraic quadruple $(E,V,\phi,\xi)$ such that $f$ is the associated morphism

$$\mathbb{P}^1\to\mathbb{P}^1.$$

This conjecture asserts the missing converse in the correspondence between the geometric data of algebraic quadruples and morphisms of projective lines. The source does not indicate whether the conjecture is known or remains open.