Will a CTO (or CTB) Gel improve the CRI of a light source

Someone is going to come along with better 'numbers' for this, but imagine the Histogram of a picture of a clown in a flower garden.
Then add a slope [filter] to that histogram.
You still have a very wobbly line, but with a slope to it.

I can only see a non-flat light emitter being the same in practise.

I spent all last week filming in a real courtroom with real 'office lighting'. They replaced all the lighting with good LED panels, even though it involved building an entire new perspex diffuser ceiling to accommodate it. They wouldn't have done that if they could have corrected for the 'office' lights.

It depends on why the source has a low CRI... is it due to one measurement being extremely far off, is it due to many measures being notably off, or is it due to all measures being somewhat off?

If it was a single measurement being relatively high (or range centered around a single max value) then a filter could theoretically attenuate that spectrum if designed for it specifically. But knocking off the high points also reduces light intensity/availability. You cannot add color/light to a light source with a filter; you can only absorb/redirect.

Filters don't add anything, they only attenuate (reduce) part of what is already there more than they reduce other parts of what is already there.

If the light source has a low CRI due to missing portions of the visible spectrum, then no filter will improve that.

If the light source has a low CRI due to one single narrow part of the visible spectrum being much brighter than the rest, then a filter that attenuates that narrow band might be helpful (at the cost of total usable light). But that's not the situation with most low quality light sources, and that's not what CTO and CTB filters do, either.

When visually evaluating color prints, we always wanted a light source with a high CRI. We could purchase light booths and lamps. Initially, Kodak recommended the use of a 40W fluorescent tube Cool White mounted together with a 60W tungsten standard household bulb. Photofinishing inspection tables used several pairs. The combination delivers a CRI in the high nineties. This works!

You might improve it a bit, but you shouldn't bother.

CRI is defined by how close the spectrum is to a black body with that temperature. Simple filters like those two in question will never actually fix any problems of the light. However, changing color temperature will usually slightly change CRI as well. This is a common trick to improve nameplate CRI slightly (in addition to tweaking of light for just 8 reference points).

Because spectrum doesn't actually get improved, your light will remain pretty horrible even if CRI seemed to get a bit higher. So, the better option is to add a coating that absorbs and re-emits part of the light, which can fix many shortcomings of the light. Generally you have fluorescence and phosphorescence as the physical mechanisms, but there are other options too - say upconverters to get blue from red light.

But it is usually much cheaper and easier to simply buy a new better bulb :)