ady exist when the age the Universe is less than $1{\,\mathrm{Gyr}}$ [see, e.g., @Fan:2001aa; @Willott:2010aa; @Mortlock:2011aa; @Venemans:2013aa; @Wu:2015aa]. This poses a question about the formation mechanism of SMBHs in such a short interval. Among the scenarios for the SMBH seed formation [see, e.g., @Volonteri:2012ab; @Haiman:2013aa for a review], including the dense stellar cluster scenario [see, e.g., @Omukai:2008aa; @Devecchi:2009aa; @Katz:2015ab; @Yajima:2016aa and reference therein], following two are the most studied: the direct collapse BH (DCBH) and the population III (Pop III) remnant BH scenarios.

In the former scenario, supermassive stars of $\sim 10^5\,M_\odot$ collapse to form seed BHs with approximately the same mass. Specifically, supermassive stars are envisaged to form in exceptional environments in the high-$z$ Universe, for example, in atomic-cooling halos where the ${\mathrm{H_2}}$ cooling is totally suppressed by very strong far ultraviolet (FUV) irradiation [e.g., @Sugimura:2014aa]. While the seed BHs in this case are rather massive with $\sim
10^5\,M_\odot$, their number density might be too small to explain all the observed high-$z$ SMBHs due to the stringent necessary conditions [@Dijkstra:2008aa; @Dijkstra:2014aa; @Agarwal:2012aa; @Sugimura:2014aa; @Sugimura:2016aa; @Inayoshi:2015ab; @Chon:2016aa].

In the latter scenario, the remnant BHs of Pop III stars [@Yoshida:2008aa; @Hosokawa:2011aa; @Hosokawa:2016aa] are thought as SMBH seeds [@Alvarez:2009aa; @Jeon:2012aa]. Contrary to the DCBH scenario, they are abundant but the problem is whether they can actually grow to the SMBHs from smaller initial mass of $\lesssim 10^3\,M_\odot$ [@Susa:2014aa; @Hirano:2015aa] within the available time. Although BHs can acquire the mass by collisions with other BHs [@Tanikawa:2011aa], the BH collisions often result in ejection of the merged BHs from the host halo due to the recoil of gravitational wave emission [e.g., @Baker:2006aa; @Koppitz:2007aa]. Thus, the feasibility of this scenario relies on