Uniform Vaught's conjecture

Let $\phi$ be an $\mathcal{L}_{\omega_{1},\omega}$-sentence of quantifier depth $\alpha$, and let Scott rank refer to the Scott rank of a model of $\phi$. Uniform Vaught's conjecture. There is a function

$$f \colon \omega_{1} \to \omega_{1}$$

such that whenever $\phi$ has a model of Scott rank greater than $f(\alpha)$, it has continuum many models of Scott rank at most $f(\alpha)$. This is a uniform strengthening of Vaught's conjecture concerning the number and Scott ranks of models of infinitary sentences; the supplied passage gives no evidence of a resolution.