Packing two graphs of even girth 10

Two graphs G1 and G2 on n vertices are said to pack if there exist injective mappings of their vertex sets into [n] such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollobás and Eldridge and, independently, Catlin, asserts that, if (Δ(G1)+1)(Δ(G2)+1)≤n+1, then G1 and G2 pack. We consider the validity of this assertion under the additional assumptions that neither G1 nor G2 contain a 4-, 6- or 8-cycle, and that Δ(G1) or Δ(G2) is large enough (≥940060).