Most popular CFD codes like OpenFOAM or ANSYS-FLUENT use finite-volume discretizations on collocated unstructured meshes. While these discretizations are preferred over staggered ones for their simplicity, some errors may arise due to this approach, such as the checkerboard problem. A possible solution to this problem would be using a compact Laplacian operator $\lapld$ instead of a wide-stencil Laplacian operator $\lapldc$. However, even for symmetry-preserving discretizations \cite{TRI08-JCP}, this approach may lead to instabilities for highly distorted meshes due to the (artificial) kinetic energy added, which is given by: \begin{equation} \label{presTerm} \presh^T (\dive \mathbf{\tilde{A}}^{-1}\gradd - \dive_c \mathbf{A}^{-1}\graddc) \presh = \presh^T (\lapld - \lapldc) \presh . \end{equation} This work gives the necessary and sufficient conditions, for explicit or implicit time integration, to remove these instabilities by maintaining $\lapld - \lapldc$ small and semi-negative definite. These results are summarized in the following theorem: \textbf{Theorem} Assume our projection method preserves the symmetry of the differential operators and ends up with a velocity correction that adds a kinetic energy error of the form $\presh^T (\lapld - \lapldc) \presh$ (such as the Fractional Step Method or PISO); then, this contribution is negative if and only if the volume-weighted interpolator is used and these two geometrical conditions are satisfied by the mesh: \begin{itemize} \item \textbf{1.} $V_k = \sum_f \tilde{V}_{k,f}n_{i,f}^2, \forall k\in\{1,...,n\}, i\in\{x,y,z\}$, \item \textbf{2.} $\sum_f \tilde{V}_{k,f}n_{i,f}n_{j,f}=0, \forall k\in\{1,...,n\}, i,j\in\{x,y,z\}, i\neq j $. \end{itemize} where $\tilde{V}_{k,f}$ is the quantity of staggered volume associated to control volume $k$, and $n_{i,f}$ are the components of the face-normal vector. This imposes geometrical constraints in the mesh that will be discussed during the conference. \bibitem{TRI08-JCP} F.X. Trias, O. Lehmkuhl, A. Oliva, C.D. P\'{e}rez-Segarra, R.W.C.P. Verstappen. Symmetry-preserving discretization of Navier-Stokes equations on collocated unstructured meshes. Journal of Computational Physics, 258:246-267, 2014.