QCD corrections, the soft-gluon resummation (threshold resummation) has been performed at next-to-next-to-leading logarithmic (NNLL) accuracy for the total cross section and invariant mass distribution, modifying the total cross section further by a small amount if the central scales are chosen as $\mu_R=\mu_F=Q/2$ [@Shao:2013bz; @deFlorian:2015moa]. Very recently, the N$^3$LO QCD corrections have been computed in the Born-improved HTL resulting in a small modification of the cross section beyond NNLO[@Banerjee:2018lfq; @Chen:2019lzz; @Chen:2019fhs; @Spira:2016zna]. These N$^3$LO QCD corrections in the HTL have been merged with the full top-mass effects of the NLO calculation [@Chen:2019fhs]. The calculations in the HTL have been improved by several steps including mass effects partially at NLO. The full mass effects in the real correction terms $d_{ij}$ have been included by means of the full one-loop real matrix elements for $gg\to HHg, gq \to HHq, q\bar q\to
HHg$. This improvement reduces the Born-improved HTL prediction for the total cross section by about 10% [@Frederix:2014hta; @Maltoni:2014eza] and is called the “FTapprox” approximation. The calculation of the full real matrix elements has been performed by using the [MG5\_aMC@NLO]{} framework [@Alwall:2014hca; @Hirschi:2015iia]. Another improvement has been achieved by an asymptotic large-top-mass expansion of the full NLO corrections at the level of the 