  In this note, we are concerned with one particular class of fully nonlinear
Hessian quotient type equations, which were initiated by Chu and Jiao in
[Curvature estimates for a class of Hessian type equations. Calc. Var. Partial
Differential Equations $\textbf{60}$ (2021).]. Previously, various second order
estimates have been derived for the $(k,l)$ quotients when $k-l \geq 2$, by
following arguments of Chu-Jiao. Here we settle the remaining $k-l=1$ case by
invoking a convenient lemma of Chen-Dong-Han in [Interior Hessian estimates for
a class of Hessian type equations. Calc. Var. Partial Differential Equations
$\textbf{62}$ (2023).].
