can this underwater photo be saved?

I am more familiar with Photoshop and Paint Shop Pro so need to generalise a bit. But the same principle applies whatever photo editor you use. I have scuba dived a lot and when I did I owned a Nikonos underwater camera so am familiar with this issues of underwater photography.

There are a few things to do with every photo - more so with those like this one that have specific problems due to the environment where they were shot. The adjustments are - adjust contrast/ brightness, adjust color and adjust sharpness and noise levels.
As to color, I would be inclined to do that first as if you can get the photo looking half way decent by adjusting color this will assist when you come to make other adjustments. In this case there is a strong blue cast so you need to use the sliders in your color adjustments to reduce the blue / cyan levels in this photo. You could also tweak the reds up a bit as these tend to get filtered out in sea water. Be a bit cirumspect with yellows. The fish need some yellow as this is their base color but if you tweak this too much up it can make the water an unpleasant urine color (yuk)
As to contrast and brightness, adjust the global / normal contrast in the photo till it looks stronger as contrast is very suppressed due to the diffused blue lighting and presence of water and particles in the water. Then use the clarity slider to adjust micro contrast till it looks better. That slider is great in any image editor as it helps apparent sharpness as well. If it is too dark after this then adjust brightness to get this looking good. Those three things should help tremendously. Finally use your sharpness and noise sliders.
You will have a much better looking photo after this - how good depends on your judgment. Experiment but try smaller adjustments first. Remember the basic issues in this photo are the blue cast introduced by water which needs the blues reduced and reds increased plus the loss of contrast due to the diffused lighting and particulate matter in the water.

Afterthought. You may find that after making the above adjustments the tiny particles in the water are more apparent. In Photoshop and Paint Shop Pro there are "noise" filters some of which are designed specifically to remove small imperfections like that. Not sure about LR although the digital camera noise filter may help (Camera noise is different to what I am talking about though. But if you can, I would consider it.

John, I did a little o this, little o that, and got this.
Auto levels, fade a bit, auto color, fade a bit, heal brush on top white areas, unsharp mask a bit, selective color add a little black to whites, run through neat image, auto contrast, dupe background and blend with softlight, fade a bit, mask top half with quick mask and gradient, auto levels, fade a bit, curves adjustment. I may have gone overboard though

Geez, sorry John, I am an idiot and didn't even read that you're only using LR. I did this in PS. You couldn't really follow my lead in LR. I''ll try it there. OK, I'm back, I can't offer you much for LR, this is the kind of image I would take into PS. Sorry about that.

In lightroom up the blacks and jack up the contrast, maybe increase clarity.

that gives you this, I think it needs more than that

Photoshop would be much easier - curves layer, set black point on black stripe and white point on the whitest fin gets you most of the way there. Hard to do in lightroom but here is my attempt. First I warmed it up using white balance then boosted exposure until the histogram was shifted to the right edge. Then boosted black until histogram was a bit past the left edge. Boost contrast, clarity and vibrance then sharpen and noise reduction. Still not enough red, though...

Quickly in photoshop with a curve layer to set black and white points, slightly drop the reds and lighten. De-noise and USM.

Image...