Does Beefeater Gin improve with age?

Nope

Gin is just another type of distilled grain alcohol (moonshine, vodka, ...) with juniper and other aromatic flavors infused. Age in the bottle is unlikely to change the ethanol molecules, and any reactions occurring in the flavoring will also be minimal and a matter of taste.

Distilled spirits that do improve with 'age' (Bourbon, whiskey, cognac, ...) are aged in wood casks or barrels, and over the years compounds leached from the wood darken and flavor the liquor. These are also not likely to improve further once moved into glass bottles for retail sale.

It has been my experience, however, that the longer I drink gin on any given night, the better and better it gets!