Does $\mathcal{P}\neq\mathcal{NP}$ imply $\mathcal{NP}\neq\mathcal{NP}^{\mathcal{NP}}$?

I cannot see whether the implication in my question title holds. To me, it sounds reasonable to assume that, but I cannot find any source stating that this is true, false, or unknown at the moment.

So, does $$\mathcal{P}\neq\mathcal{NP}$$ imply $$\mathcal{NP}\neq\mathcal{NP}^{\mathcal{NP}}$$?

You are asking whether $$\text{P} \neq \text{NP}$$ implies that the polynomial hierarchy $$(\text{PH})$$ does not collapse to its first level.
The class $$\text{NP}^\text{NP}$$ is also known as $$\Sigma_2^P$$ which is one of the classes on the second level of the polynomial hierarchy.