In our paper [Bernoulli 26(2), 2020, 1381-1409], we found all strong Markov solutions that spend zero time at $0$ of the Stratonovich stochastic differential equation $d X=|X|^α\circ dB$, $α\in (0,1)$. These solutions have the form $X_t^θ=F(B^θ_t)$, where $F(x)=\frac{1}{1-α}|x|^{1/(1-α)}\text{sign}\, x$ and $B^θ$ is the skew Brownian motion with skewness parameter $θ\in [-1,1]$ starting at $F^{-1}(X_0)$. In this paper we show how an addition of small external additive noise $\varepsilon W$ restores uniqueness. In the limit as $\varepsilon\to 0$, we recover heterogeneous diffusion corresponding to the physically symmetric case $θ=0$.