Cantelli's conjecture on Gaussian mixtures

Let $X,Y$ be independent real random variables with standard Gaussian distribution, and let $\varphi$ be a measurable non-negative function. The random variable $X+\varphi(X)\cdot Y$ is said to have a Gaussian distribution when its law is Gaussian.

Cantelli's conjecture. The random variable $X+\varphi(X)\cdot Y$ has a Gaussian distribution if and only if $\varphi$ is constant.

This question asks whether a non-constant positive measurable function can preserve Gaussianity in this construction. The paper's abstract states that it constructs a counterexample, so the conjecture is refuted.