A further remark concerning Nagel cevians (Q914154)

The Nagel cevians \(n_ a\), \(n_ b\), \(n_ c\) of a triangle ABC are the lines \(AN_ a\), \(BN_ b\), \(CN_ c\), where \(N_ a\), \(N_ b\), \(N_ c\) are the points of contact of the three excircles of the triangle with BC, CA, AB respectively. It is known that \(\Sigma n_ a\leq 9r+L(R-2r),\) where R and r are the circumradius and inradius of the triangle. The question is: what is the minimum value of L ? The author shows that \(6.258998...\leq L_{\min}\leq 6.478487,\) and states that there is a strong evidence that \(L_{\min}=6.258998..\).