Regularity conjecture for the Hofstadter H sequence limit density

Let $mu$ be the absolutely continuous limit measure of the numbers $\alpha H(n)\mod 1$, where $\alpha$ is the real root of $x^3+x=1$ and $H$ is Hofstadter's $H$ sequence. Regularity conjecture. The probability density function of $mu$ is continuous but not differentiable. The paper notes that the rate of decay of the Fourier coefficients is unknown; the asserted regularity is presented as a consequence of the observed decay and remains unproved.