Solvability conjecture for positive Hessian quotient equations

Let $M$ be a connected, closed Riemannian manifold with metric $g$, and let $\chi$ be a symmetric $(2,0)$ tensor on $M$. Suppose that there is an admissible function $v\in C^4(M)$ in the sense of Definition

. For $1\leq l\leq n-1$, let $F=\frac{\sigma_n}{\sigma_l}$ be the positive Hessian quotient operator. **Solvability conjecture.** The equation

is solvable for any smooth right-hand side, up to a multiplicative constant. This conjecture proposes a general solvability result for positive Hessian quotient equations on closed Riemannian manifolds; the supplied text does not state whether it has been proved or disproved.