Can you hedge older T-Bonds with futures?

I was told that the exchange sets a conversion factor that remains fixed during the futures contract "life".

However the exchange fixes the factor for simplicity, altough it varies.
 
The conversion factor is fixed for a given bond in the deliverable basket for the life of a given futures contract. The conversion factor doesn't change. If you think about what the conversion factor is and how it is computed, you'll realize that it has to remain constant for each given bond/contract combination.
 
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