What is the conjectured relationship between P (PTime) and Type 1 (context-sensitive) languages?
If $\mathrm{P\subseteq CSL}$, then $\mathrm{P\subseteq DSPACE}(n^2)$. By a padding argument, this implies
$$\mathrm{DTIME}(t(n))\subseteq\mathrm{DSPACE}\bigl(t(n)^\epsilon\bigr)$$
for every superpolynomial well-behaved function $t(n)$ and every $\epsilon&gt;0$. I believe such a strong advantage of space over time is not expected to be true. The best currently known result in this direction is
$$\mathrm{DTIME}(t(n))\subseteq\mathrm{DSPACE}(t(n)/\log t(n)),$$
due to Hopcroft, Paul, and Valiant.