Existence of solution to modified Gursky-Streets equation

We solve the modified Gursky-Streets equation, which is a fully nonlinear equation arising in conformal geometry with uniform $C^{1,1}$ estimates when (i) $\gamma >0$ and $1\le k\le n$ or (ii) $r>0$ and $2sk\le rn$. We also prove the existence of a Lipschitz continuous viscosity solution when $r\mathrm{\ne }0$.