Candidates’ positions on Brexit are, naturally, under most scrutiny. None favours a second referendum—that would play terribly with the party’s grassroots. とはどういう意味ですか?

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"[The] Candidates' positions on Brexit are, naturally, under [the] most scrutiny."
This means that the candidates have many positions (opinions,) but the ones about Brexit are under the most scrutiny.
When something is "under scrutiny," it is being carefully examined and criticized.

"None favours a second referendum—that would play terribly with the party's grassroots."
Basically, nobody wants a second referendum. A referendum is basically when a group of people is invited to vote on something.
"That would play terribly with the party's grassroots." I think this means that it would go against their most basic values.