  This work studies Liouville conformal blocks of irregular type with the
insertion of at least one level-$3$ degenerate field admitting a Fibonacci
fusion rule. We algebraically derive the corresponding third-order BPZ
equations for regular blocks and their modifications when a rank one irregular
operator is inserted. Employing Lefschetz thimbles as integration cycles, we
then successively proceed to construct integral representations and prove that
they satisfy the corresponding BPZ equations. Finally, we show that taking a
semiclassical limit, these integral representations can be expressed in terms
of Heun functions and have correct leading behaviors consistent with conformal
weights and fusion rules.
