Conjecture about reversal operations on strings (with duplicates)

Here is a check in Python that your proposed counterexample is indeed a counterexample. I took the subflip function from @AleksejsFomins's answer.

import itertools

def subflip(s, i, j):
    return s[:i] + s[i:j][::-1] + s[j:]

s = "22131441"
t = "41211342"

flipped_strings = set([s])
num_flips = 0

while True:
    if t in flipped_strings:
        print("Minimal number of flips:", num_flips)
        break

    new_flipped_strings = set()
    for i, j in itertools.combinations(range(len(s) + 1), 2):
        new_flipped_strings.update(subflip(f, i, j) for f in flipped_strings)

    flipped_strings = new_flipped_strings
    num_flips += 1

The output is (near instantaneously):

Minimal number of flips: 4

And indeed, if you prepend the 4 on both strings this script confirms that three flips suffice to change one string into the other.

I have run a numerical simulation on your proposed counter-example, and the minimal number of flips for the string without prefix is 4, which is more than 3

CODE (Python3)

def subflip(s, i, j):
    return s[:i] + s[i:j][::-1] + s[j:]

s1 = "22131441"
s2 = "41211342"
N = len(s1)

def test(s1, s2, pref, depth):
    for i in range(N-1):
        for j in range(i+2, N+1):
            s1_a = subflip(s1, i, j)
            pref_a = pref+[(i, j)]
            if s1_a == s2:
                print(s1_a, s2)
                print("Win", pref_a)
                return True

            if depth - 1 > 0:
                if test(s1_a, s2, pref_a, depth-1):
                    return True
    return False

test(s1, s2, [], 4)

Answer:

Win [(0, 4), (0, 6), (3, 8), (4, 7)]

Edit: This is a depth-first search, so I ran the code several times for increasing value of depth until I got an answer

Note: I do not believe I deserve the bounty, because you proposed the counter-example yourself, I just checked it