CANBERRA, Australia –  Australia’s beleaguered prime minister faces a political test next month as an opinion poll shows his center-right party faces a large swing against it in a by-election in the nation’s most conservative state.

Prime Minister Tony Abbott’s ruling coalition is battling poor polling and internal division over whether Australia should lift a ban on same-sex marriage.

A poll published in The Australian newspaper on Tuesday shows that Abbott’s Liberal Party could lose a by-election on Sept. 19 in Western Australia state.

The poll showed the Liberal Party was more popular than the opposition Labor Party by 51-to-49 percent. That result was within the poll’s 4.3 percentage point margin of error. It was based on a weekend survey of 508 voters.