The common monodromy-invariant cycle conjecture for glued K3 fibrations

Let $p=(F_{\beta}\colon X\to [0,1])\in U_{HSVZ}$ be a glued fibration of a K3 surface to the segment, and suppose that $F_{\beta}$ factors through the two Lagrangian fibrations $\pi'$ and $\pi”$, with fiber classes $e'$ and $e”$. Then there is a $1$-homology class, denoted $e'\cap e”$, such that

Common cycle conjecture. The class $e'\cap e”$ is primitive in both $H_{1}(e',\mathbb{Z})$ and $H_{1}(e”,\mathbb{Z})$, and is monodromy invariant with respect to both $\pi'$ and $\pi”$.

This conjecture concerns the compatibility of the two Lagrangian fibrations arising near the Satake boundary of the moduli space. The supplied text states the expected primitive and monodromy-invariance properties but gives no evidence of a resolution.