MMSE transform injectivity conjecture up to reflection

Let $X$ and $Z$ be zero-mean random variables, and for $\rho\in[0,\infty)$ let ${\mathsf{mmse}}(X,\rho)$ denote the minimum mean-square error in estimating $X$ from a Gaussian-noise observation at signal-to-noise ratio $\rho$. MMSE transform injectivity conjecture. For all ${\mathsf{snr}}\in[0,\infty)$,

$${\mathsf{mmse}}(X,{\mathsf{snr}})={\mathsf{mmse}}(Z,{\mathsf{snr}})$$

if and only if $X$ is identically distributed as either $Z$ or $-Z$. This asks whether the MMSE transform determines a zero-mean input distribution up to reflection; the source presents it as an interesting question that remains to be answered.