A model scenario has recently been introduced by de la Reza and colleagues to explain the presence of very strong Li lines in the spectra of some low-mass K giant stars. In this scenario all ordinary, Li-poor, K giants become Li rich during a short time (~105 yr) when compared to the red giant phase of 5 x 107 yr. In this "Li period," a large number of the stars are associated with an expanding thin circumstellar shell supposedly triggered by an abrupt internal mixing mechanism resulting in a surface new 7Li enrichment. This Letter presents nearly 40 Li-rich K giants known up to now. The distribution of these Li-rich giants, along with 41 other observed K giants that have shells but are not Li rich, in a color-color IRAS diagram confirms this scenario, which indicates, also as a new result, that a rapid Li depletion takes place on a timescale of between ~103 and 105 yr. This model explains the problem of the presence of K giants with far-infrared excesses presented by Zuckerman and colleagues. Other present and future tests of this scenario are briefly discussed.