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I'd say ask an engineer...
40mm solid beech is not going to flex much over 300mm, but the units only really support at the sides, right? or is there a rail at the back against the wall? I feel like fixing down to a rail against the wall would make a difference, also if you could add a rail along the top/front of the units but might not be space for that.
Is it block laminated? There could be risk of it splitting I guess, depending how much you load the overhang. 300mm doesn't seem that controversial though.
Breakfast bar overhang of 300mm in 40mm thick solid beech worktop. Will I get away with this or should I add a supporting leg? (The overhang juts out from a 600mm deep base unit).