A simple observation on random matrices with continuous diagonal entries

Let $T$ be an $n\times n$ random matrix, such that each diagonal entry $T_{i,i}$ is a continuous random variable, independent from all the other entries of $T$. Then for every $n\times n$ matrix $A$ and every $t\ge0$ $$\p\Big[|\det(A+T)|^{1/n}\le t\Big]\le2bnt,$$ where $b>0$ is a uniform upper bound on the densities of $T_{i,i}$.