Generalized Borel conjecture for aleph-zero-bounded groups

Let $(G,\odot)$ be an $\aleph_0$-bounded topological group, and let $X\subseteq G$ be a Rothberger bounded subset. Let the weight of $G$ denote the least cardinality of a base for its topology.

Generalized Borel conjecture. The cardinality of $X$ is no larger than the weight of $G$:

$$|X|\leq w(G).$$

This is the proposed generalization of Borel's Conjecture from the real numbers to $\aleph_0$-bounded topological groups. The source does not specify the resolution status here.