C2 estimate conjecture for general Hessian quotient equations

Let $1\leq l<k$ and consider the general Hessian quotient equation

$$\frac{\sigma_k}{\sigma_l}(g_u)=f.$$

Let $u$ be a solution on a closed Riemannian manifold $(M,g)$ endowed with a symmetric $(2,0)$ tensor $\chi$. C2 estimate conjecture. One should have

$$\|u\|_{C^2(M)}\leq C\bigl(1+\|u\|_{C^0(M)}\bigr),$$

where $C$ depends only on $n$, $k$, $l$, $\chi$, $f$, and $g$. This is presented as the expected a priori estimate for general Hessian quotient equations once a $C^0$ estimate is available; the supplied text gives no evidence that it has been proved or disproved.