Indeed. Some Muslim countries like Bahrain, Iran, and Qatar are below replacement level, and the UAE is far below replacement level. So we can rule out any strong version of this claim, e.g. that Muslim culture automatically ensures high fertility. The question is thus whether any weaker version of this claim is correct.

I suspect that it is not. Tajikistan, for instance, banned the hijab back in 2017. In February 2004, Turkmenistan banned men from wearing long hair and beards for anti-Islamist reasons, although that regulation has been said to have been loosening over time. Uzbekistan defeated a Taliban-friendly Islamist insurgency of its own.

These countries (excluding Kyrgyzstan) have usually been much closer to following the authoritarian, nativist, and personalist path, although Kazakhstan and Uzbekistan seem to have moved on a liberal path since the departures from power of Nazarbayev and Karimov. That leaves Tajikistan and Turkmenistan as the more 'based' countries of Central Asia, at least insofar as governments are concerned.

In Tajikistan's case, I suspect that the persistence of nativism is responsible. Tajikistan has been just as anti-Russian as anti-Muslim, for example, President Emomali Rahmon's name was Emomali Rahmonov until he had the last two letters removed for nativist reasons, and, in April 2016, Tajik parents were banned from giving Russian-style names to their newborn children. Basically, Tajikistan's governmental policies have moved the country somewhat closer to those of an ethnostate.

One reason that I suspect that Islam is not responsible is that Turkmenistan is probably the least anti-Muslim country of all five Central Asian countries. For instance, it was one of the few countries around the world to have been on good terms with the Taliban. But Turkmenistan's demographic trends are actually rather unimpressive compared to those in the less Islamic Kazakhstan in particular. Should Islam be the determinant of these trends, Turkmenistan should really be the most impressive of the five rather than the least, and Kazakhstan should really be the least impressive of the five rather than the most.

In Kazakhstan's case, this miracle of sorts clearly happened during Nazarbayev's long rule. Since I think that nativism is more likely to be the determinant, I do wonder if Nazarbayev was more of a nativist than I thought.