Fast answer Yes,and no damage.your truck will just put juice in,if its not taken it won"t be a problem for the truck, It is what its supposed to do, if one cranks the crap out of a bat in cold weather and really runs them down and it starts just before dying, The truck will charge em,and kinda quickly too, a good bat will charge itself if left alone for awhile, to a point, it wants to take a charge... They are for all sound reasons good bats or they would not have taken the punishment, If U are about to take a very long ride, u would do it, But are U sure they will Hold the charge? if U stop and they don"t re start your truck is it worth it, or is the wear and tear and fuel worth charging bats also, They should be brought up to a full charge at a slower amp preferably but not really the most criticle thing in the world, and THEN wait one FULL day to test em, the little ball floats are fine as long as U do it on a rested battery OR U will get a false reading,and also if the ball things are new or dried out ,,not used in a long time , fill em and let em sit for a few minues with the acid...and then use em...checking a resting battery meaning after a full charge has been taken and waiting 24 hours for the battery to rest is the only way one will get a true reading testing any battery... or again U will get a false reading,The desulphaters are the way to go, but the bat has to sit for a minimum of 72 hours for a mostly good bat, and any bat that sits for more than 4 days should be on a maintainer or its life will be shorter than that of a maintained bat, that is when guys say my bats are crap ,they need maintanence too..Hope u do not mind a little more than asked for....Randy