。
She refuses to bring up the topic.
Literal
She [topic-は] that story [to-に] try-to-touch [contrast-は] do-not.
話にふれる literally 'to touch on the story' is the standard way to say 'broach' or 'bring up' a topic in conversation. The construction ~ようとはしない ('won't try to') is stronger than a plain negative — it implies stubborn unwillingness rather than mere absence of action, with the contrastive は emphasizing the gap between expectation and behavior ('she won't even try'). Choosing the polite ~ません instead of ~ない gives the line an observational, slightly distancing tone, as if the speaker is reporting a pattern they've noticed.