Yes, but he doesn't need another equipment grounding conductor other than the one with his 4 wire feeder.
He does not have a cable assembly, he has to add the 4th conductor, if it were EMT, Rigid, or IMC, the metallic raceway is a code compliant grounding conductor, with steel flex, it is not as the overcurrent protection is over 20A & I assume will be over 6 feet in length. See 250.118(5).
250.118 Types of Equipment Grounding Conductors.
The equipment grounding conductor run with or enclosing
the circuit conductors shall be one or more or a combination
of the following:
(1) A copper, aluminum, or copper-clad aluminum conductor.
This conductor shall be solid or stranded; insulated,
covered, or bare; and in the form of a wire or
a busbar of any shape.
(2) Rigid metal conduit.
(3) Intermediate metal conduit.
(4) Electrical metallic tubing.
(5) Listed flexible metal conduit meeting all the following
conditions:
a. The conduit is terminated in listed fittings.
b. The circuit conductors contained in the conduit are
protected by overcurrent devices rated at 20 amperes
or less.
c. The combined length of flexible metal conduit and
flexible metallic tubing and liquidtight flexible metal
conduit in the same ground-fault current path does
not exceed 1.8 m (6 ft).
d. If used to connect equipment where flexibility is necessary
to minimize the transmission of vibration from
equipment or to provide flexibility for equipment that
requires movement after installation, an equipment
grounding conductor shall be installed.
Underline by me.