Indications that strengthen the conjecture: NEXP ⊊ EXP^NP
Being pulled down by the gravitational force from $\mathrm{BPP}$, hardly any convincing evidence can be lurking on the surface of $\mathrm{EXP}^{\mathrm{NP}}$ for you to see.

Only after proving $\mathrm{BPP}\subseteq \mathrm{P}^\mathrm{NP}$, this class would give up on $\mathrm{EXP}^\mathrm{NP}$. But then, $\mathrm{EXP}$ and $\mathrm{NEXP}$ would have their chances to thrive (on their own).