Boyd's conductor 30 Mahler measure conjectures

Let $F(b,c)$ denote the lattice sums used in the paper, and let $m(\alpha)$ and $g(\alpha)$ be the Mahler measures defined by

$$m(\alpha)=\int_0^1\int_0^1\log\left|y+y^{-1}+z+z^{-1}+\alpha\right|\,\mathrm d\theta_1\,\mathrm d\theta_2,$$

and

$$g(\alpha)=\int_0^1\int_0^1\log\left|(y+1)(z+1)(y+z)-\alpha yz\right|\,\mathrm d\theta_1\,\mathrm d\theta_2,$$

where $y=e^{2\pi i\theta_1}$ and $z=e^{2\pi i\theta_2}$. Boyd's conductor 30 Mahler measure conjectures. The following formulas are numerically true:

$$\frac{15}{4\pi^2}F(2,15)=3m(1)-g(3),$$ $$\frac{15}{\pi^2}F\left(2,\frac{5}{3}\right)=3m(1)+g(3).$$

These formulas arise from an unproven conjectural relation between the conductor $30$ elliptic curve and $g(3)$, together with a modular-equation relation among the lattice sums. The source provides numerical evidence but no proof or resolution.