Does jazz "ax(e)" derive from jazz "chops"?

[Gerald Cohen]

    I'm preparing an article on slang "ax(e)" (= musical instrument),
with full gratitude and credit to all the ads-l contributors to this
topic of two years ago; I'll also contact each one individually.

    But I have one main point of uncertainty: whether to connect this
"ax(e)" with slang "chops" (jazz term denoting musical ability or
skill). This "chops" seems readily derivable from jazz "chops" (= "a
trumpet player's strength of embouchure"--HDAS), which in turn
derives from "chops" (= mouth or lips).
So perhaps "ax(e)" (= musical instrument) derived from the differing
meanings of "chops": 1) mouth; skill in playing e.g., a trumpet, 2)
strikes with an axe.

    The very knowledgeable members of this list often see things that
I miss.  The above semantic hypothesis (axe as something one chops
with) seems plausible to me, but am I perhaps missing something here?
The inclusion of the "chops" discussion is just an addendum to the
"ax(e)" treatment and could easily be removed if it presents problems.

    For ready access I present HDAS information after my signoff.

Gerald Cohen

HDAS says of ax(e) meaning #4: "sense development unknown; perh.
sugg. by _sax_; perhaps infl. by _swing_. Jazz: a musical instrument,
esp. one on which jazz or rock music is played, as a saxophone,
trumpet, or guitar."
The first attestation cited is from 1955.

   Meanwhile, HDAS says of _chops_:
meaning #1a: "Now esp. Black English: the mouth or lips;..." -- first
attestation cited: 1589.
  #2a: "Jazz. a trumpet player's strength of embouchure" -- first
attestation cited: 1947.
#2b. "Orig. Jazz: musical ability or skill; (hence) skill or ability
of any sort." -- first attestation cited: 1968.