Similar questions have been asked, but I've not seen one that matches the situation described below.
I have 3 cats. One is 5 years old and has been with us most of that time, one is 2 years old and we've had her since she was a tiny kitten and the third is around 16 months old and we've had her around a year.
We have never had any issues with the older 2 cats and socializing, but the third cat is proving to be extremely challenging in this regard.
From the day we got her she was focused 100% on hiding (not uncommon for cats in my experience) however a year or so later and she still hides all the time.
We have noticed she will quite happily come out when we are asleep or out of the house (we have a camera set up) and she is perfectly happy with the other 2 cats, (they will curl up together, etc.)
We started by trying the "let her have her space" tactic; however, this caused issues when she stayed under a chair for over a week, only coming out when she was really hungry or thirsty.
We then tried blocking all of the "Hidey holes" except for a couple (such as a "cat cave" from a cat tree) to try to bring her out of the shell; however, she would just do the same and stay in constantly.
We have got to the point now where occasionally we can coax her out from a hiding spot onto the top of a bed or sofa; however, she shows fear and submission when we approach her. We occasionally try to start fussing her once she has relaxed a little (she will stay on the sofa if I sit on it as well but it takes over 30 minutes for her to stop showing fear).
Once we start fussing she becomes a different cat and will roll onto her belly, slow blink, purr like mad, lick our hands and rub her head against us.
We have tried suggestions from the vets such as Feliway, different approaches, removal from the other cats, we have watched a myriad of youtube videos on this, with no success.
Essentially my question is: how can we get the cat to start to come out of her shell a bit more (we aren't expecting miracles, some cats are naturally more scaredy-cat like), so that we can interact with her more?
Note - we contacted the previous owner who said that she had been the "runt" of the litter and essentially rejected by her siblings and mother (there were a lot of them when we picked her up), so there may be a great hangover from her kitten hood.
Note 2 - even if this doesn't improve its not the end of the world, she is still a special cat who is a part of our family and if that's how she feels comfortable that's fine, it would just be nice for us as owners
Thanks in advance.