  Structural properties of CuAl$_{2}$O$_{4}$, which was recently argued to show
unusual suppression of the Jahn-Teller distortions by the spin-orbit coupling,
are investigated under pressures up to 6 GPa. Analysis of X-ray powder
diffraction experiments shows that CuAl$_{2}$O$_{4}$ gets unstable and
decomposes onto CuO and Al$_{2}$O$_{3}$ at pressures $\sim$ 6 GPa and
temperature $\sim$ 1000 K. This finding is complemented by the DFT+U+SOC
calculations, which demonstrate that this instability is partially driven by a
(relatively) large compressibility of strongly Jahn-Teller distorted CuO.
