Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers

For totally positive algebraic integers $\alpha \ne 0,1$ of degree $d(\alpha )$, we consider the set $\mathcal{L}$ of values of $M(\alpha )^{\frac{1}{d(\alpha )}}=\Omega (\alpha )$, where $M(\alpha )$ is the Mahler measure of $\alpha$. C. J. Smyth has found the four smallest values of $\mathcal{L}$ and conjectured that the fifth point is $\Omega ((2\cos \frac{2\pi }{60})^2)$. We prove that this is so and, moreover, we give the sixth point of $\mathcal{L}$.