> Randomized studies show that men now face more hiring discrimination than women do
This is an odd phrasing. These are exclusive categories that cover the possibility space†; it makes sense to say that "women are favored over men", but it doesn't make sense to say "men face more discrimination than women". Any number you come up with for "discrimination against men" is necessarily defined relative to the outcomes for women; you can't assign cardinal numbers to both groups.
† Not quite. The possibility space also includes children. They face far, far more discrimination than either men or women do. For example, hiring them is a serious crime.
> Gender discrimination is often regarded as an important driver of women’s disadvantage in the labour market, yet earlier studies show mixed results. However, because different studies employ different research designs, the estimates of discrimination cannot be compared across countries. By utilizing data from the first harmonized comparative field experiment on gender discrimination in hiring in six countries, we can directly compare employers’ callbacks to fictitious male and female applicants. The countries included vary in a number of key institutional, economic, and cultural dimensions, yet we found no sign of discrimination against women. This cross-national finding constitutes an important and robust piece of evidence. Second, we found discrimination against men in Germany, the Netherlands, Spain, and the UK, and no discrimination against men in Norway and the United States. However, in the pooled data the gender gradient hardly differs across countries. Our findings suggest that although employers operate in quite different institutional contexts, they regard female applicants as more suitable for jobs in female-dominated occupations, ceteris paribus, while we find no evidence that they regard male applicants as more suitable anywhere.
> Male applicants were about half as likely as female applicants to receive a positive employer response in female-dominated occupations. For male-dominated and mixed occupations we found no significant differences in positive employer responses between male and female applicants.
> both scientists and laypeople overestimated the continuation of bias against female candidates. Instead, selection bias in favor of male over female candidates was eliminated and, if anything, slightly reversed in sign starting in 2009 for mixed-gender and male-stereotypical jobs in our sample. Forecasters further failed to anticipate that discrimination against male candidates for stereotypically female jobs would remain stable across the decades.
This is a coherent claim, but it can't be summarized as "men face more discrimination than women do". Neither (1) nor (2) is an amount of discrimination.
It could be the case, for example, that there are twenty times as many jobs in male-dominated industries as in female-dominated industries, and that men and women apply to these in perfect proportion to their availability.
(Your more specific claim, women do not face any negative discrimination in male-dominated industries, will mean that the amount of negative discrimination faced by women is lower than that faced by men regardless, but this isn't a necessary part of the way you've constructed the question.)
This is an odd phrasing. These are exclusive categories that cover the possibility space†; it makes sense to say that "women are favored over men", but it doesn't make sense to say "men face more discrimination than women". Any number you come up with for "discrimination against men" is necessarily defined relative to the outcomes for women; you can't assign cardinal numbers to both groups.
† Not quite. The possibility space also includes children. They face far, far more discrimination than either men or women do. For example, hiring them is a serious crime.