  A bijection between $(31245,32145,31254,32154)$-avoiding permutations and
$(31425,32415,31524,32514)$-avoiding permutations is constructed, which
preserves five classical set-valued statistics. Combining with two codings of
permutations due respectively to Baril--Vajnovszki and Martinez--Savage proves
an enumerative conjecture posed by Gao and Kitaev. Moreover, the generating
function for the common counting sequence is proved to be algebraic.
