The most common and most likely answer to this crossword clue is the 7 letter word MAESTRO
I've seen this crossword clue in Independent
M
A
E
S
T
R
O
More clues leading to the result MAESTRO
- Accomplished musician excited Tom's ear
- Band backed trio appearing without one conductor
- Some rat misused funding system?
- Rattle, say, very nearly storms out without a note
- Great conductor? Most are excited
- Big Eddy's out of line and ultimately lacking genius
- Report seaman holding expert up
- Making comeback as alternative band recruits singular top musician
- Staff run out to welcome English conductor
- Most are blown for conductor, perhaps
Buddy explains!

The straight part
We know that "eminent conductor" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case eminent conductor = maestro.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 MAESTRO mean?
MAESTRO noun- A master in some art, especially a composer or conductor.
- A gang elder in prison.