The most common and most likely answer to this crossword clue is the 8 letter word BLINDEST
I've seen this crossword clue in Financial Times
B
L
I
N
D
E
S
T
Buddy explains!

The straight part
We know that "most insensitive" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case most insensitive = blindest.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 BLINDEST mean?
BLINDEST verb- To make temporarily or permanently blind.
- To curse.
- To darken; to obscure to the eye or understanding; to conceal.
- To cover with a thin coating of sand and fine gravel, for example a road newly paved, in order that the joints between the stones may be filled.
- (of a person or animal) Unable to see, due to physiological or neurological factors.
- (of an eye) Unable to be used to see, due to physiological or neurological factors.
- Failing to see, acknowledge, perceive.
- Of a place, having little or no visibility.
- Closed at one end; having a dead end
- Having no openings for light or passage.
- (in certain phrases) Smallest or slightest.
- Without any prior knowledge.
- Unconditional; without regard to evidence, logic, reality, accidental mistakes, extenuating circumstances, etc.
- Unintelligible or illegible.
- Abortive; failing to produce flowers or fruit.