a $\varphi_{k,\epsilon}$ which satisfies pointwise a.e. in $(-1,2T) \times B_{4R}$ $$\label{eq:kthEllipticProb}
\partial_{t}\varphi_{k,\epsilon}+\mathcal{M}^{+}_{\frac{\lambda}{4},\Lambda}(D^{2}\varphi_{k,\epsilon})=-1_{A_{r_{k-1},r_{k+2}}}(\Div u_{\epsilon})_{-}, \quad \quad \quad \varphi_{k,\epsilon} \geq \frac{1}{2p}.$$ We now collect several regularity estimates for $\varphi_{k,\epsilon}$. As these will depend on $L^{q}_{t,x}$ norms of $\Div u_{\epsilon}$, let us first bound this quantity uniformly in $\epsilon$. We choose $\epsilon_{0}$ depending on $u$ and $\gamma$ such that for each $k \geq 2$, $$[\max(r_{k-1}-\epsilon_{0},0),r_{k+2}+\epsilon_{0}] \subset [r_{k-2},r_{k+3}].$$ Recalling that $\Div u_{\epsilon}=\Div u * \eta_{\epsilon}$ where $\eta_{\epsilon}$ is supported in $B_{\epsilon}$ and using Jensen’s inequality, we find for $\epsilon<\epsilon_{0}$ $$\begin{aligned}
\|(\Div u_{\epsilon})_{-}\|_{L^{q}([0,T] \times A_{r_{k-1},r_{k+2}})}&\leq \|(\Div u)_{-}*\eta_{\epsilon}\|_{L^{q}([0,T] \times A_{r_{k-1},r_{k+2}})} \\
&\leq \|(\Div u)_{-}\|_{L^{q}([0,T] \times A_{\max(r_{k-1}-\epsilon,0),r_{k+2}+\epsilon})} \\
&\leq \|(\Div u)_{-}\|_{L^{q}([0,T] \times A_{r_{k-2},r_{k+3}})}\leq 5 \gamma,\end{aligned}$$ where we used in the last step. By the maximum principle (see for instance Theorem 2.4 in [@MR1789919]) and it now follows that $$\|\varphi_{k,\epsilon}\|_{L^{\infty}([-1,2T] \times B_{4R})} \leq \frac{1}{2p} + C\gamma.$$ For reasons which will be apparent in Step 4, we choose $\gamma$ such that $C\gamma<(2p)^{-1}$ to ensure $$\label{eq:maxPrinc}
\|\varphi_{k,\epsilon}\|_{L^{\infty}([-1,2T] \times B_{4R})} \leq \frac{1}{p}.$$ Furthermore, we have the following interior estimates, following from Theorem 9.1 of [@MR1789919] $$\begin{aligned}
\label{eq:intEst}
\|\nabla_{x} \varphi_{k,\epsilon} \|_{L^{\infty}([0,T] \times B_{3R})}+\|\{\partial_{t},\nabla^{2}_{x}\}\varphi_{k,\epsilon}\|_{L^{1}([0,T] \times B_{3R})}\leq M_{p}^{1}.\end{aligned}$$ *Step 4: Sub-solution construction*\
We now define a collection of cutoff functions $\{\eta_{k}\}_{k=1}^{N}$ as follows. First, 