The focus of this work is on LHs, which locally mix EXX and semi-local exchangeenergy densities in coordinate space to model nondynamical correlation (NDC) in a flexible way, governed by a so-called local mixing function a(r) (LMF). We expect, however, that the developments reported in this work can also be extended to RSLHs, and potentially even to rung 5 functionals. A wide variety of LMFs has been proposed, and we refer the reader to our 2019 review for an overview, and to refs. 23, 24 for more recently proposed models. We note in passing that strong-correlation corrections also modify the LMF in important ways, but we will explicitly not examine such terms in this work. A particularly useful expression of an LH functional is shown in eq. 1, where we can identify full exact exchange on the left, a crucial middle NDC term, and a largely dynamical correlation (DC) contribution on the right. We refer to a recent detailed discussion regarding the interpretation of the middle term as an NDC contribution. Its integrand multiplies the complement of the LMF, 1 -a(r), by the difference between semi-local exchange and EXX energy densities, e sl X (r) and e ex X (r), respectively. It can contain a so-called calibration function (CF), G(r), to mitigate the so-called gauge problem of LHs that is due to the ambiguity of exchange-energy densities. However, we have recently shown that the gauge problem can also be suppressed by a suitable form of LMF. Indeed, we will show that this extends also to the type of LMFs proposed in this work, obviating the need of a CF.