Sweden’s inconclusive September election saw a record 161 women take their seats in the 349-member Riksdag – the highest proportion of female MPs in Europe (ahead of two other Nordic countries, Finland and Norway) and the seventh highest in the world. But the country that in 2014 proudly declared that it had “the first feminist government in the world” has never had a female prime minister, and is about to enter an unprecedented fifth month under a caretaker administration.

The key to resolving the latter problem could well lie with 35-year-old Annie Lööf, a former business minister and, since 2011, the youngest ever leader of the tax-cutting, business-promoting, immigrant-welcoming Centre party. Neither is it inconceivable that Lööf, Sweden’s most trusted politician in a host of polls in 2017 and 2018, could also find herself rectifying the former problem – although most analysts think it unlikely and she herself has said the top job is not her focus.

The election left the two dominant centre-right and centre-left blocs separated by a single seat, deadlocked – and facing a major problem in the form of the far-right, anti-immigration Sweden Democrats, the country’s third biggest party. Lööf’s Centre, the fourth largest party with 31 seats, a sharp increase on its tally in the previous parliament, is part of the four-member centre-right Alliance, and has sworn never to be a part of, or support, a government backed by the Sweden Democrats.

Centre’s votes could give a majority to Stefan Löfven, the outgoing Social Democrat prime minister and leader of the three-party centre-left bloc – except that Lööf campaigned on a promise not to govern with the Social Democrats either.

Both Löfven and the Alliance leader, Ulf Kristersson, have tried and failed to form a new government. Lööf was also asked to explore coalition options, but soon gave up, blaming the Social Democrats’ reluctance to accept liberal reforms.

Only two more formal attempts to form a coalition are allowed before fresh elections must be called on 23 January. If Löfven tacks right and wins Lööf’s support, she could demand high office as a reward, perhaps even very high office.

Whether she would get it, of course, is another matter.

* This article was amended on 14 January 2019. An earlier version referred to Lööf as “Sweden’s most trusted politician in a host of polls since 2017”. That should have referred to polls in 2017 and 2018. This has been corrected.