A topological space has *countable spread* if every discrete subspace is at most countable.

By Theorem 8.10 in Todorcevic's book "Partition Problems in Topology", PFA implies that each regular space $X$ with countable spread is hereditarily Lindelof and hence has countable pseudocharacter (which means that each singleton $\{x\}$ in $X$ is a $G_\delta$-set). This result entails the following

**Corollary.** *Under PFA, each topological group of countable spread has countable pseudocharacter.*

Problem.Is this corollary true in ZFC? In other words:has each topological group of countable spread countable pseudocharacter?

**Remark.** The Peng-Wu example of an L-group shows that topological groups with countable spread need not be separable.