Regularity of linear Bellman equation

Question

Let $f(x,t):B_1 \times [0,1] \to \mathbb{R}$ be Lipschitz function on both $x$ and $t$, $\varphi$ be Lipschitz function on both $x$ and $t$ on the parabolic boundary of $B_1 \times [0,1]$. Let $A$ be a countable set. Suppose $a_{ij}^\nu(x,t)$, $a_i^\nu(x,t)$ are regular enough, we consider the linear Bellman equation, $$ u_t(x,t) +\sup_{\nu \in A} \Big(a_{ij}^\nu(x,t) u_{x_ix_j}(x,t) +a_i^\nu(x,t) u_{x_i}(x,t) \Big)=f(x,t)$$ on $ B_1 \times [0,1]$ with parabolic boundary value $\varphi$. I am looking for the interior $C^{2,1}$ regularity of the solution. I think it is true, but I cannot find a proper reference. There is a close one in Theorem 14.15 "Second Order Parabolic Differential Equations" by Lieberman, Gary M., see also the comments for the linear operator $L_\nu$ just below the theorem. It states quite ambiguously in that comment that $f(x,t)$ should be differentiable on both $x,t$, but I think it is not necessary.