One thing that is actually correct is that there's more variation in gene expression between men and women. On of the reasons for this is because, chromosomally, women have two X chromosomes, which pair up, and then randomly one or another turns off. These means that the gene expression is mixed between both parents.
On the other hand, the Y chromosome is largely only effective at turning off gene expression. You can see how there's less redundancy here.
But intelligence isn't a gene. Researchers have, since the time of Dalton, tried to find a simple, biological basis for genius. You know, memory capacity, reaction times, brain size, brain structure convolution, etc. They haven't found anything -- literally everything has turned out to be a false start, even brain size, which, has been shown, within families does not even predict g.
In the mid ranges, there are greater standard deviations, yes. But every single normed test is normed on a sample on the order of 1000. They're designed for regular people. The designer of the Weschler has adamantly opposed the use of IQ tests for anything other than clinical settings, for this reason. It's just no good at drawing conclusions on the extremes of ability.
A better guide might be actual performance. The IOI this year had more girls than ever before -- 11. That's nearly 300% more than last year, where they had 4, and one medal. They are:
Emina Bukva (Bosnia and Herzegovina) Constanza Contreras (Chile) Anna Currel (Spain) Romina Huenchunao (Chile) Vaiva Imbrasaite (Lithuania) Taksapaun Kittiakrastien (Thailand) [Silver] Sepideh Mahabadi (Iran) [Gold!] Radwa Metwali (Egypt) Katie O'Mahony (Ireland) Phitchaya Phothilimthana (Thailand) [Silver] Ye Wang (USA) [Silver]
Sepideh Mahabadi had one of the best performances of anyone. If you're familiar with IOI scoring, only the top 1/12 are to get golds, and 2/12 to get silvers, thus 1 gold and 3 silvers out of 11 implies that they did at least as good as the boys, and in fact somewhat better.
Were the 'standard deviation' explanation correct they should have, instead, had 0.4 girls earning maybe 0.01 medals. It just doesn't work. Psychometrics are a non-science.