Is it possible to improve the present LAGEOS-LAGEOS II Lense-Thirring experiment?

The Lense-Thirring effect is currently being measured by means of a combination of the orbital residuals of the nodes of LAGEOS and LAGEOS 11 and the perigee of LAGEOS II. The claimed total error should be of the order of 20-25%. The most insidious systematic error is due to the mismodelled even zonal harmonics of the geopotential and amounts to 12.9%, according to the full covariance matrix of the EGM96 Earth gravity model up to degree l = 20. The role and the importance of the LAGEOS LAGEOS II Lense-Thirring experiment is investigated. Using other suitable combinations with orbital elements of the other existing laser-ranged satellites does not yield significant improvements except, perhaps, for one combination including the nodes of LAGEOS, LAGEOS II and Ajisai and the perigee of LAGEOS II. The related systematic error due to the mismodelled even zonal part of the geopotential reduces to almost 10.7%, according to the full covariance matrix of EGM96 up to degree l = 20.