Connection formulas relating Frobenius solutions of linear ODEs at different
Fuchsian singular points can be expressed in terms of the large order
asymptotics of the corresponding power series. We demonstrate that for the
usual, confluent and reduced confluent Heun equation, the series expansion of
the relevant asymptotic amplitude in a suitable parameter can be systematically
computed to arbitrary order. This allows to check a recent conjecture of
Bonelli-Iossa-Panea Lichtig-Tanzini expressing the Heun connection matrix in
terms of quasiclassical Virasoro conformal blocks.