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I have never seen any manager I worked before he requested us to log/write down the lessons learned even the one who requested never shared or discussed.
Most probably because they don't expect to do the same project again for the same company or at least not during their tenure with the company. Also Lessons Learnt can be subjective to time hence what is relevant today in the Lessons Learnt might not be of much use1 or2 years down the line with changes in the industry and economics.