Homotopical constancy conjecture for non-splitting singularities

Let $(X_0,f_0)$ be an isolated singularity and $(X_t,f_t)$ a one-parameter deformation. Suppose that $(X_0,f_0)$ does not split, meaning that the critical set of $f_t$ has only one point in a sufficiently small neighbourhood for sufficiently small $t$. Consider the Milnor fibration of the isolated singularity of $(X_t,f_t)$.

Homotopical constancy conjecture. If the isolated singularity $(X_0,f_0)$ does not split, then the Milnor fibration of the isolated singularity of $(X_t,f_t)$ is homotopically constant for $t$ close to $0$.

This predicts topological stability of the Milnor fibration under deformations that do not split the singularity. The source presents the assertion as a question and gives no resolution.