Is the mode elicitable relative to unimodal distributions?

Statistical functionals are called elicitable if there exists a loss or scoring function under which the functional is optimal point forecast in expectation. While mean and quantiles elicitable, it has been shown Heinrich (2014) that mode cannot be elicited true distribution can follow any Lebesgue density. We strengthen this result substantially, showing with continuous density unique local maximum. Likewise, fails to identifiable relative class.