The insulin feedback loop is based on free fatty acids not ketones, so
the answer is no.  But there is some degree of linkage over a period of
time.  The natural sequence in week 3 is to reduce the production of
ketones so it very closely matches the BMR.  During Induction the body
burns far more fat than needed, and ketones pass through the skin and
kidneys and intestines and leak out everywhere.  After Induction the
sticks go down towards trace because the production of ketones starts
to very closely match the metabolic usage.  The presence of the trace
is from the fact that ketones can't be stopped from passing through the
kidneys into the urine.  They just dissolve through the cells.  And the
smallness of the trace is from the fact that the body has adjusted its
production of ketones to match the metabolic demand.  Just how much
insulin has to do with this feedback adjustment, I don't know, but I
doubt the effect is completely zero.

On the insulin feedback loop and the free fatty acids.  This is to
respond to a meal with high fat content.  Digestion breaks fat into
glycerol and fatty acids for absorbtion into the blood stream.  Then
the liver and/or fat cells take the glycerol and the fatty acids and
they stitch them back into fat.  Most likely the fatty acids get
editted for length int he process.  So eat enough fat and your insulin
will go up just enough to keep the fatty acids handled.  *OR* release
enough stored fat and your insulin will go up just enough to keep the
fatty acids handled.  Interesting stuff, that.  It imples that as long
as your blood sugar is low, and your metabolism isn't broken by insulin
resistance and so on from diabetes, the level of fatty acids in the
blood *can't* get dangerously high.  And since the supply of ketones
in the blood is based on the liver's supply of free fatty acids, that's
why folks who see dark test strips during Induction don't go into
ketoacidosis and die.  There's an indirect insulin feebback loop that
keep the levels from becoming toxic no matter from dietary fat or
withdrawn stored fat or both in combination.

This also explains two other mechanisms one anti-obvious and one obvious.

The anti-obvious one is that with a reasonable total calorie intake,
eating more dietary fat results in more stored fat being withdrawn from
storage.  It's why the original fat fast experiment showed the folks
eating 1000 calories of 90% fat lost more than the folks eating 0
calories.  It's why Atkins is high-fat, medium-protein not the other
way around.  For the same total calories, more fat annd less protein
works better all the way until you hit your daily protein minimum and
trigger starvation mode.  It is very much the opposite that eating more
fat should lead to loss, but as long as you keep your total calories
limited it does and you'll see post after puzzling post reporting that
someone ate a huge T-bone and had a whoosh, or that someone binged by
drinking over a cup of heavy cream and had a whoosh.  The hormone
mechanism is that eating dietary fat indirectly triggers the release
of glucagon, and glucagon levels determine flow of fat out of storage.
More dietary fat, more glucagon, more fat withdrawn from storage.

The obvious one is that if you eat far too much calories in the form of
far too much fat, it overwhelms the glucagon effect.  But it takes far
above the guideline calories to do so and no plan endorses chronic
over eating so no plan endorses eating that much anything whether it's
fat or not.  But there's the biochemical mechanism raising it's complex
head: Eat too much fat and the intestines release too much free fatty
acids.  Fatty acids *indirectly* cause glucagon release and increase
stored fat withdrawal, but too much fatty acids *directly* cause insulin
release.  Since insulin in the blood suppresses glucagon production,
the overwhelm situation starts forcing fat back into storage anyways.

So go from 60-80 grams of fat (low) to 100-140 grams of fat (high) and
you will withdraw *more* fat from storage.  But go from 100-140 grams
of fat (high) to 200-260 grams of fat (ultra-high)and it is enough to
overwhelm the loop, trigger the extra insulin, and push fat into
storage.  Atkins is high-fat and takes advantage of this mechanism, but
Atkins is not over-eat so it avoids the overwhelm trigger.  It
combines the anti-obvious of more-fat-more-withdrawn and the obvious
of eat-too-much-store-new-fat.