I believe this is false.
A program found counterexample on $8$ vertices and exhaustive search confirmed it.
The edges are:
[(0, 4), (0, 5), (0, 6), (1, 4), (1, 5), (1, 7), (2, 4), (2, 6), (2, 7), (3, 5), (3, 6), (3, 7)]
It is $3$-connected.
There are other counterexamples (modulo errors).