The quote titling this post - "Progesterone hormone treatment significantly increased the risk of ASD [autism spectrum disorder]" - comes from the findings reported by Michael Davidovitch and colleagues .
Researchers set out to examine whether there was any *association* between the receipt of infertility treatments and offspring risk of a diagnosis of autism spectrum disorder (ASD). They concluded that use of IVF treatment - in vitro fertilisation, where a woman's egg(s) are removed and fertilised outside of the body then 'put back in' - probably doesn't show any significantly enhanced risk for offspring autism. When however it came to the use of progesterone hormone treatment complementing such assisted reproductive methods, there may be something to see...
Before going further into these findings, I want to hark back to some discussions (see here) on the previous research in this area. The paper by Liang Liu and colleagues  meta-analysing the peer-reviewed literature on the use of assisted reproductive technology (ART) and offspring risk of autism did suggest there might be something more to see; albeit concluding that: "The complexity of ART treatment renders the identification of individual risk factors extremely challenging." I say this on the basis that various different 'pregnancy and birth' factors have been *correlated* with offspring autism risk down the years (see here for example) and teasing apart which ones are the more 'important' is difficult to say the least.
The Davidovitch study relied on data for over 100,000 children (males) born between 1999 and 2008. Authors identified approaching one thousand children diagnosed with an ASD representing about 1% of the total cohort. As well as looking at IVF, they also examined "five hormone treatments" pertinent to ART, singling out progesterone hormone treatment as part of their observations.
There is lots more work to be done in this area before too many sweeping generalisations are made. Aside from the issue of teasing out exactly what conception/pregnancy variables *might* be related to offspring risk of autism (including reproduction itself!) there is a question of mechanism(s) to also consider. The authors refer to the term "epigenetic modification by progesterone" but only few details are provided. I might also at this point introduce a recent piece from Jill Escher  who provided some really interesting commentary on how synthetic hormone drugs *might* have played a role in her own family circumstances as a template for some further studies. I say this bearing in mind that progesterone should not be too readily confused with progestins .
To close, I also read some other work from Michael Davidovitch recently  talking about cell (mobile) phone use and "the development of joint attention in infants" pertinent to "the development of autistic features among a vulnerable subgroup of infants." I'm slightly less enthralled about such suggestions and the strength of any 'technology use cause autism' sentiments but am willing to keep an open mind on how our fascination with mobile phones *might* impact on parent-child interactions (but not necessarily in the context of autism)...
 Davidovitch M. et al. Infertility treatments during pregnancy and the risk of autism spectrum disorder in the offspring. Progress in Neuro-Psychopharmacology and Biological Psychiatry. 2018. June 1.
 Liu L. et al. Association between assisted reproductive technology and the risk of autism spectrum disorders in the offspring: a meta-analysis. Scientific Reports. 2017; 7: 46207.
 Escher J. Bugs in the program: can pregnancy drugs and smoking disturb molecular reprogramming of the fetal germline, increasing heritable risk for autism and neurodevelopmental disorders? Environ Epigenet. 2018 Apr 26;4(2):dvy001.
 Spark MJ. Progesterone or progestogen or progestin; which is it? BMJ. 2009; 339: b4380.
 Davidovitch M. et al. The Role of Cellular Phone Usage by Parents in the Increase in ASD Occurrence A Hypothetical Framework. Medical Hypotheses. 2018. June 7.