entropy and adopting some ideas from [@LedrappierYoung1985], Feng and Hu proved that for any contracting $C^1$ conformal IFS, the push-forward measure of every ergodic invariant measure under the coding map is exact dimensional, with dimension given by the projection entropy divided by the Lyapunov exponent. Later this result was further extended to some random self-similar measures [@FalconerJin2014; @SagliettiShmerkinSolomyak2018] and push-forward measures of ergodic invariant measures for some random conformal IFSs [@MihailescuUrbanski2016]. It is worth pointing out that the exact dimensionality of overlapping self-similar measures was first claimed by Ledrappier; nevertheless no proof has been written out (cf. [@PeresSolomyak2000 p. 1619]). This property was also conjectured later by Fan, Lau and Rao in [@FanLauRao2002]. The first result for affine IFSs is due to Bedford [@Bedford1984] and McMullen [@McMullen1984], who independently calculated the Hausdorff and box-counting dimensions of a special class of planar self-affine sets (which are now called Bedford-McMullen carpets) and showed that they are usually different. McMullen [@McMullen1984] also implicitly proved the exact dimensionality of self-affine measures on the Bedford-McMullen carpets, and calculated the precise value of the dimension. Later, Gatzouras and Lalley [@GatzourasLalley1992] and Barański [@Baranski2007] obtained similar results for a class of more general carpet-like self-affine sets in 