FPT vs W[P] - Parameterized Complexity In parametrized complexity, $\mathsf{FPT} \subseteq \mathsf{W}[1]$ $\subseteq \mathsf{W}[2]$ $\subseteq \ldots \subseteq \mathsf{W}[P]$. It is conjectured that each of the containments is proper.

If $\mathsf{FPT}=\mathsf{W}[P]$ then $\mathsf{P}=\mathsf{W}[P]$.

But does it follow that 


If $\mathsf{FPT}=\mathsf{W}[1]$ then $\mathsf{FPT}=\mathsf{W}[P]$ ?
or
If $\mathsf{W}[t-1]=\mathsf{W}[t]$ (for some t) then $\mathsf{FPT}=\mathsf{W}[P]$ ?
This question is tricky as the answer (as far as I know) is still "don't know".

To add some weight to this, Flum &amp; Grohe [1] give as open problems (p. 164):


  
  Is the $\mathrm{W}$-hierarchy strict under the assumption $\mathrm{FPT} \neq \mathrm{W[P]}$?
  For $t \geq 1$, does the equality $\mathrm{W}[t] = \mathrm{W}[t + 1]$ imply $\mathrm{W}[t] = \mathrm{W}[t + 2]$?
  


Moreover, in Downey and Fellow's recent monograph [2] the strongest (outright) statement they make is (p. 521):


  A more subtle hypothesis is that the $\mathrm{W}$-hierarchy is proper, and, in particular, $\mathrm{W}[1] \neq \mathrm{W}[2]$.


There is no following (or later) statement along the lines "otherwise the $\mathrm{W}$-hierarchy would collapse", or similar.

This is also preceded by:


  A weaker hypothesis might be that for some $t$,
  $$\mathrm{FPT} \neq \mathrm{W}[t]$$


Implying that it is possible that $\mathrm{FPT} = \mathrm{W}[t-1]$ with no other effects on the hierarchy.

References:


J. Flum and M. Grohe, "Parameterized Complexity Theory", Springer, 2006.
R. Downey and M. Fellows, "Fundamentals of Parameterized Complexity Theory", Springer, 2014.