Consider a log-correlated Gaussian field Γ and its associated imaginary multiplicative chaos :e^{i \beta \Gamma}: where β is a real parameter. In [AJJ22], we showed that for any nonzero test function f, the law of \int f :e^{i \beta \Gamma}: possesses a smooth density with respect to Lebesgue measure on C. In this note, we show that this density is strictly positive everywhere on C. Our simple and direct strategy could be useful for studying other functionals on Gaussian spaces.
