Consider the family of fully non-linear equations (1.7) with respect to the degenerating background metrics $\omega_{t}$, $t\in(0,1]$. Assume that we have solutions $\varphi_{t}\in C^{2}$, normalized by $\sup_{X}\varphi_{t}=0$. Assume that $\lambda[h_{t,\varphi_{t}}]\in\Gamma_{k}$, for some fixed $k$, $1\leq k\leq n$. Fix $q>n/k$. Then $\|\varphi_{t}\|_{L^{\infty}}$ is uniformly bounded by a constant $C$ depending only on $n,k,q$ $\omega_{X},\chi$ $\|e^{nF_{t}}\|_{L^{q}(\omega_{X}^{n})}$ and $\frac{c_{t}^{n}}{V_{t}}$.