Expansion conjecture for Seifert fibered spaces over the 2-sphere

Let $Y$ be a Seifert fibered space over $S^2$ with $\varepsilon(Y)>0$. An expansion is the operation described in the paper, and a sequence of expansions may be empty. For integers and rational parameters as displayed, write

$$Y'=S^2\left(1;\frac{p_1}{q_1},\ldots,\frac{p_l}{q_l}\right).$$

Expansion conjecture. The space $Y$ smoothly embeds in $S^4$ if and only if it is obtained by a possibly empty sequence of expansions from some $Y'$ of the displayed form, with

$$\frac{p_i}{q_i}>1$$

for every $i$, such that $Y'$ also smoothly embeds in $S^4$.

This would reduce the embedding problem for Seifert fibered spaces over $S^2$ with positive normalized Euler invariant to the subclass of partitionable spaces and their expansions. The preceding discussion gives constructions in one direction and explains that the conjecture would exclude the unresolved family with a complementary class of size three, but no resolution is supplied here.