Stolarsky bounds for the invariant logarithmic mean

Let $L(a,b)=\dfrac{b-a}{\log b-\log a}$ be the logarithmic mean, let $L_{3}$ denote its invariant mean in three variables, and let $U_{0}$ and $U_{1}$ be Stolarsky's three-variable means defined by

$$U_{0}(a,b,c)=\left(\dfrac{1}{2}\dfrac{(a-c)(-c+b)(-b+a)}{a\ln b-a\ln c+(\ln a)c-(\ln b)c-(\ln a)b+(\ln c)b}\right)^{1/2},$$ $$U_{1}(a,b,c)=\dfrac{1}{2}\dfrac{(b-c)(a-c)(a-b)}{a(b-c)\log a-b(a-c)\log b+c(a-b)\log c}.$$

Stolarsky's bounds conjecture. For all $(a,b,c)\in R_{+}^{3}$,

$$U_{0}(a,b,c)\leq L_{3}(a,b,c)\leq U_{1}(a,b,c),$$

with equality if and only if $a=b=c$.

These inequalities would provide tighter lower and upper bounds for the invariant logarithmic mean than the basic geometric and arithmetic mean bounds. The source reports strong evidence but does not establish the claim.