Ahsanallah ilaikum “It is permissible for the menstruating woman, the postpartum woman, and even the one in janābah to recite the Qur’an and touch the Muṣḥaf” Someone said this, is the latter true? And they also said it is permissible to touch mushaf w/o barrier for the one in minor impurity

Ustadh AbdulAziz al-Haqqan

You can't touch the mushaf without purity unless you are using a barrier like cloth. As for reciting the Qur’an from memorization it is allowed on the correct saying except when you are upon major impurity (janabah).