Lorraine is correct, do not underestimate the ability of a BSJF to escape through any opening! My first BSJF jumped through eggcrate, second through eggcrate covered with 1/2" netting, third got out by shifting a panel of eggcrate with dual layers of 1/2" netting just enough to squeeze through a gap! If you watch your tank in the early a.m. hours, you may well see the BSJF swimming around up to with its head sticking out of the water looking for any possible way out. Forget about fairy wrasses, blennies, gobies, etc. I have lots of these and haven't lost one to carpet surfing -- escape potential of the jawfish is an order of magnitude higher!
I've read that once a burrow is established the risk of jumping dramatically decreases. Maybe so, but I will not be taking any chances. Jawfish in my tank = escape-proof cover in place.
Also in my opinion/experience (with a perimeter brace tank in a niche with wall or cabinetry along the front and back) having no cover is preferable to any cover that is not 100% escape proof. It is likely one or more of my previous BSJF would be alive if the tank were uncovered; sure they would have still jumped but probably would have bounced off the lights back into the water, or landed on the perimeter brace and squirmed back into the tank. Instead, they were caught above the eggcrate and became jerky.
Using eggcrate vertically, rather than horizontally, and/or a "tent" made of 1/4" or 3/8" netting should be considered if you have a canopy that will conceal it.
I have some (hard to find) 1/4" clear mesh that I will use to create my next top when time allows.
BSJF#4 has been in my tank for about six weeks now; no doubt he has tried to jump many times. Current cover is eggcrate panels with the centers cut away (to maximize light penetration) with dual layers of 1/2" bird netting sourced from Home Depot. At each end of the tank the panels are weighted down by heavy 5-fan Propeller Breeze units from GHL/Profilux. Where the panels meet they are attached by wire ties to keep them from separating. Where the sump and chiller returns enter the top, there is a second layer of eggcrate also. This provides good aeration and minimizes light attenuation. It is a pain to remove and replace but only takes a few minutes, and I don't loose sleep at night... no more waking up in the morning and looking at the eggcrate for jerky if the BSJF is not in his burrow!
Of course as luck would have it he does have some "condition" with patchy areas of white/grey discoloration on his body. This has been going on for several weeks now. He also seems to have an intermittent rectal prolapse

His (I'm pretty sure he is a male) breathing and behavior remain normal and he is eating well. I have not tried to capture/treat him in a hospital tank but have added nitrofurazone and metronidazole to the food in case of parasite or bacterial infection. The nitrates in my tank did creep up (maybe up to 25) and I have read somewhere that jawfish skin and slime do not tolerate nitrates well. So I have been doing large frequent water changes and initated AZNO3 treatments with a decelerated dosing schedule as I have clams and a rose anenome. I'm trying to reduce feeding also (hard to do with 28 fat fish including four anthias, a dwarf fuzzy lionfish, and a lot of wrasses, three tangs, and a rabbifish!) mainly by thoroughly washing frozen foods and using more flake, pellet, and freeze dried food.
I've thought about adding Praziquantel or piperazine to the food in case of his condition being brought on by some type of tubellarian or other worm infection but have resisted so far since he doing okay and I do have coco worms and feather dusters in the tank, which could be adversely affected by anti-flatworm medications. Ich does exist in my tank -- as I believe it exists in most -- but my "canary" i.e. the juvenile blue hippo tang -- looks fine so doubt crypto is a major issue at this time.
To end this long post with a happy note, a LFS has had a pretty BSJF since 11/6/2007 at a good price ($99) and it seemed healthy. Pretty sure it is a female. We bought her home yesterday and introduced her into the tank. She hid in the rocks most of the day, but came out to eat at night and this morning we found her in the same borrow as BSJF#4! Would have gotten a picture but only the actinics were on and didn't want to freak them out with a flash from the camera. Jasper (BSJF#4) did eventually push her out but she has been intermittently hanging out about six inches away without any other signs of aggression. Hopefully they will be a couple -- if not as male/female then maybe female/female

I would have expected lots of fighting if they were both male...
Will try to update my thread in the Bluespot Club with pictures later today or tomorrow.