The temperature distribution of field Li-rich red giants suggests the presence of a population of Li-rich red clump ( RC ) stars .
One proposed explanation for this population is that all stars with masses near 2 M _ { \sun } experience a short-lived phase of Li-richness at the onset of core He-burning .
Many of these stars have low ^ { 12 } C/ ^ { 13 } C , a signature of deep mixing that is presumably associated with the Li regeneration .
To test this purported mechanism of Li enrichment , we measured abundances in 38 RC stars and 6 red giant branch ( RGB ) stars in four open clusters selected to have RC masses near 2 M _ { \sun } .
We find six Li-rich stars ( A ( Li ) \geq 1.50 dex ) of which only two may be RC stars .
None of the RC stars have Li exceeding the levels observed in the RGB stars , but given the brevity of the suggested Li-rich phase and the modest sample size , it is probable that stars with larger Li-enrichments were missed simply by chance .
However , we find very few stars in our sample with low ^ { 12 } C/ ^ { 13 } C. Such low ^ { 12 } C/ ^ { 13 } C , seen in many field Li-rich stars , should persist even after lithium has returned to normal low levels .
Thus , if Li synthesis during the He flash occurs , it is a rare , but potentially long-lived occurrence rather than a short-lived phase for all stars .
We estimate a conservative upper limit of the fraction of stars going through a Li-rich phase to be < 47 \% , based on stars that have low ^ { 12 } C/ ^ { 13 } C for their observed A ( Li ) .