The solenoid is not closed. It should be in the region of 0.001~0.005V. Switch it off if you think it is on.
Problem, how many amps is your converter good for?
It may be too much in this this case.
Solder two wires onto a tail light bulb and wire that across those posts of the solenoid and watch it for a few days. Batteries that dead can take a very long time to recharge. When the bank voltage gets to about 11, try firing that contactor (solenoid) now. If it doesn't go solid try a little longer.
If they were good deep cycle batteries, they may recover to 80~90% - if you are luck and careful recharging them. Most of all, don't push them too hard.
This is the sort of thing I did on expensive boats before the depression shut us down. I was successful at recovering the bank about 75% of the time. A four year old bank..... I still give a maybe an even bet to recover.
Matt