  We report the discovery of two Li-rich giants, with A(Li) ~ 1.50, in an
analysis of a sample of 40 giants of the open cluster Trumpler 20 (with turnoff
mass ~ 1.8 Msun). The cluster was observed in the context of the Gaia-ESO
Survey. The atmospheric parameters and Li abundances were derived using
high-resolution UVES spectra. The Li abundances were corrected for nonlocal
thermodynamical equilibrium (non-LTE) effects. Only upper limits of the Li
abundance could be determined for the majority of the sample. Two giants with
detected Li turned out to be Li rich: star MG 340 has A(Li) non-LTE = 1.54 \pm
0.21 dex and star MG 591 has A(Li) non-LTE = 1.60 \pm 0.21 dex. Star MG 340 is
on average ~ 0.30 dex more rich in Li than stars of similar temperature, while
for star MG 591 this difference is on average ~ 0.80 dex. Carbon and nitrogen
abundances indicate that all stars in the sample have completed the first
dredge-up. The Li abundances in this unique sample of 40 giants in one open
cluster clearly show that extra mixing is the norm in this mass range. Giants
with Li abundances in agreement with the predictions of standard models are the
exception. To explain the two Li-rich giants, we suggest that all events of
extra mixing have been inhibited. This includes rotation-induced mixing during
the main sequence and the extra mixing at the red giant branch luminosity bump.
Such inhibition has been suggested in the literature to occur because of fossil
magnetic fields in red giants that are descendants of main-sequence Ap-type
stars.
