Let $E, X$ be path-connected and suppose I have a fibration $p\colon E\to X$ which admits a section $s$. For a generalised homology theory $\mathcal{E}_\ast$, there is a splitting $\mathcal{E}_\ast (E)\cong \mathcal{E}_\ast(X)\oplus R_\ast$ induced by $p$ and $s$.

Choosing a basepoint $x\colon \ast \to X$, we have a homotopy fibre sequence $$ \begin{array}{ccc} F & \to & E \\ \downarrow & & \downarrow \\ \ast & \to & X \end{array} $$ which implies a map $\psi\colon \mathcal{E}_\ast(F) \to \mathcal{E}_\ast(X)\oplus R_\ast$. Commutativity of the diagram implies that projecting $\psi$ to the first factor results in the zero map.

Question: is the map $\mathcal{E}_\ast (F)\to R_\ast$ an isomorphism?

This certainly seems like it should be the case, but I cannot seem to work out the argument. I am particularly interested in the case that $\mathcal{E}_\ast = \pi^\mathrm{st}_\ast$ is stable homotopy.