Filters F for which the space F +omega embeds into a box product or real lines.

It was proven in [1] that for any ultrafilter $p \in \omega^*$, the space $\{ p \} \cup \omega$ (as a subspace of $\beta \omega$) does not embed in any box product or real lines. However, it is clear that for the filter of cofinite sets $\mathcal{F}$ (the Frechét filter), the space $\{ \mathcal{F} \} \cup \omega \simeq \omega +1$ embeds in the real line. We give some facts about the following question formulated by Rodrigo Gutiérrez: Is there a combinatorial or topological property about the filters $\mathcal{F}$ for which the space $\{ \mathcal{F} \} \cup \omega$ embeds in a box product or real lines?

Reference:
[1] F. Hernandez-Hernandez and H. A. Barriga-Acosta, {On discretely generated box products}, Topology and its Applications 210 (2016), 1-7.