The most common and most likely answer to this crossword clue is the 4 letter word DEAD
I've seen this crossword clue in Independent
D
E
A
D
More clues leading to the result DEAD
- Lifeless? Exactly!
- . Extremely old man accepting fate at last
- Very large cups hold endless French water
- Extremely insensitive
- After injecting drug, parent is numb
- So uninspiring!
- Insensitive wife finally accepted by relative
- /9 Obsolete Conservative needs, er, time? Sure thing
- Very late
- Parent swallowing tablet late
Buddy explains!

The straight part
We know that "Insensitive" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case insensitive = dead.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does DEAD mean?
DEAD noun- (with "the", a demonstrative, or a possessive) Those who have died.
- (often with "the") Time when coldness, darkness, or stillness is most intense.
- (usually plural) Sterile mining waste, often present as many large rocks stacked inside the workings.
- To prevent by disabling; stop.
- To make dead; to deaden; to deprive of life, force, or vigour.
- To kill.
- No longer living.
- Figuratively, not alive; lacking life.
- (of another person) So hated that they are absolutely ignored.
- Doomed; marked for death (literally or as a hyperbole).
- Without emotion.
- Stationary; static.
- Without interest to one of the senses; dull; flat.
- Unproductive.
- (of a machine, device, or electrical circuit) Completely inactive; currently without power; without a signal.
- (of a battery) Unable to emit power, being discharged (flat) or faulty.
- Broken or inoperable.
- No longer used or required.
- Not imparting motion or power by design.
- Not in play.
- (of a golf ball) Lying so near the hole that the player is certain to hole it in the next stroke.
- (1800s) Tagged out.
- Full and complete.
- Exact.
- Experiencing pins and needles (paresthesia).
- Constructed so as not to transmit sound; soundless.
- Bringing death; deadly.
- Cut off from the rights of a citizen; deprived of the power of enjoying the rights of property.
- (often with "to") Indifferent to, no longer subject to or ruled by (sin, guilt, pleasure, etc).
- (degree) Exactly.
- (degree) Very, absolutely, extremely.
- Suddenly and completely.
- As if dead.