This question is inspired by this question Implications between $\mathsf{L}=\mathsf{P}$ and $\mathsf{NL}=\mathsf{NP}$? We do know that ${\bf L}$ could equal ${\bf NL}$ and at the same time ${\bf P}$ could be different from ${\bf NP}$. I was wondering whether or not the inequality between ${\bf L}$ and ${\bf NL}$ imply inequality between ${\bf P}$ and ${\bf NP}$. Or we just simply do not know... 
Does ${\bf L} \neq {\bf NL}$ imply ${\bf P} \neq {\bf NP}$?
complexity classes