The Essential™ (comment-free) Convenient Palindromic Quine Golf
Jelly, 17 bytes

ðṛṾṖṚŒḄø“øḄŒṚṖṾṛð


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Verfication

A string of length 17 has only 217 = 131,072 subsequences, so this can be done by brute force.

$ # Generate all 131,072 subsequences of the source code.
$ jelly eun '“ðṛṾṖṚŒḄø“øḄŒṚṖṾṛð”ṾḊṖŒPY' &gt; subsequences
$ wc -l subsequences
131072 subsequences
$ cat verify.py
import jelly

# Discard extraneous output to STDOUT (just a bunch of 0's and [0]'s).

jelly.print = lambda _, end: None

for line in open('subsequences').readlines():
        line = line.strip()
        try:
                output = jelly.stringify(jelly.jelly_eval(line, []))
        except:
                output = ''
        if output == line or output == 'ðṛṾṖṚŒḄø“øḄŒṚṖṾṛð':
                print(repr(line))

$ time python3 verify.py
'ðṛṾṖṚŒḄø“øḄŒṚṖṾṛð'

real    0m9.678s
user    0m9.653s
sys     0m0.028s


How it works

ðṛṾṖṚŒḄø“øḄŒṚṖṾṛð  Main link. No arguments. Implicit left argument: 0

       ø           Make the chain to the right niladic.
        “øḄŒṚṖṾṛð  Yield the string 'øḄŒṚṖṾṛð'.
ð                  Make the chain to the right and up to ø dyadic.
                   The main link is now a monadic 2,0-chain, so the dyadic chain
                   is called with the implicit left argument 0 and right argument
                   'øḄŒṚṖṾṛð'.
 ṛ                 Yield the right argument, i.e., 'øḄŒṚṖṾṛð'.
  Ṿ                Uneval; yield the string representation of the right argument,
                   i.e., '“øḄŒṚṖṾṛð”'.
   Ṗ               Pop; remove the last character, yielding '“øḄŒṚṖṾṛð'.
    Ṛ              Reverse, yielding 'ðṛṾṖṚŒḄø“'.
     ŒḄ            Bounce, yielding 'ðṛṾṖṚŒḄø“øḄŒṚṖṾṛð'.
                   (implicit) Print the last return value.