Logically this makes sense, but testing it would be better since not all of our spells behave logically.
BoD replaced BoA on single target fight starting here and a few following posts.
There is no reason BoD should not buff Drain Soul on execute phase :
- Soul siphon takes the number of affliction effects into account
- Death's Embrace increases damage to shadow spells on execute
- BoD is an affliction spell, same as BoA
- BoD does shadow damages, same as BoA
=> conclusion : Bod buffs execute DS the same way as BoA.