A note on the PDE approach to the $L^{\mathrm{\infty }}$ estimates for complex Hessian equations on transverse Kähler manifolds

We adapt the PDE approach of Guo-Phong-Tong and Guo-Phong-Tong-Wang [17, 18] to prove an $L^{\mathrm{\infty }}$ estimate for transverse complex Monge-Ampère equations on homologically orientable transverse Kähler manifolds. As an application, we obtain a purely PDE proof of the regularity of Calabi-Yau cone metrics on Q-Gorenstein T-varieties.