The natural isomorphism conjecture between hyperseries and surreal numbers

Let $\textbf{Hy}$ be the field of hyperseries and $\textbf{No}$ the field of surreal numbers. For each hyperseries $f(x)\in\textbf{Hy}$, consider its value $f(\omega)\in\textbf{No}$. Natural isomorphism conjecture. There is a natural isomorphism

$$\textbf{Hy}\longrightarrow\textbf{No}$$

that sends every hyperseries $f(x)$ to its value $f(\omega)$. The conjecture reflects the close structural analogy between hyperseries and surreal numbers; the source gives no resolution.