关于Oppenheim不等式的加强

      最近刘健在【http://www.icstm.ro/DOCS/josa/josa_2012_1/a_01_Liu.pdf】给出Oppenheim不等式

$R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}\geq (r_{1}+r_{2})(r_{2}+r_{3})+(r_{2}+r_{3})(r_{3}+r_{1})\\+(r_{3}+r_{1})(r_{1}+r_{2})$

一个优雅的证明，并猜测其加强版

$R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}\geq (w_{1}+w_{2})(w_{2}+w_{3})+(w_{2}+w_{3})(w_{3}+w_{1})\\+(w_{3}+w_{1})(w_{1}+w_{2})$

成立。我尝试了刘健文章中的技术路线，但是未获成功。这个加强版倒底是否成立?