Rodriguez-Villegas conjectures for logarithmic Mahler measures

For a Laurent polynomial $P\in\mathbb C[x_1^{\pm1},\ldots,x_n^{\pm1}]$, let $m(P)$ denote its logarithmic Mahler measure. Let $f_{3,15}$ and $f_{4,6}$ be the modular forms defined in the preceding Bessel-moment conjecture, and let $L(f_{3,15},4)$ and $L(f_{4,6},5)$ be their corresponding special $L$-values. Rodriguez-Villegas conjecture. The following evaluations are expected:

$$m(1+x_1+x_2+x_3+x_4)=6\left(\frac{\sqrt{15}}{2\pi}\right)^5L(f_{3,15},4),$$

and

$$m(1+x_1+x_2+x_3+x_4+x_5)=3\left(\frac{\sqrt{6}}{\pi}\right)^6L(f_{4,6},5).$$

These conjectures connect logarithmic Mahler measures with special modular-form $L$-values; the supplied source gives no resolution status.