Monotonicity inequalities sufficient for the Stolarsky bounds

Let $L(a,b)=\dfrac{b-a}{\ln b-\ln a}$ be the logarithmic mean, and let $U_{0}$ and $U_{1}$ be the three-variable means defined in the source. For positive $a,b,c$, form the three logarithmic means $L(a,c)$, $L(a,b)$, and $L(c,b)$.

Sufficient inequalities for Stolarsky's bounds. The inequalities

$$U_{0}(L(a,c),L(a,b),L(c,b))\geq U_{0}(a,b,c)$$

and

$$U_{1}(L(a,c),L(a,b),L(c,b))\leq U_{1}(a,b,c)$$

for all $(a,b,c)\in R_{+}^{3}$ would suffice to prove the bounds conjectured for $L_{3}$.

This is presented as a sufficient route to proving the preceding bounds, rather than as an independently named conjecture; the source does not state whether these inequalities are known or open.