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We prove that the generalized Carleson operator $C_d$ with polynomial phasefunction is of strong type $(p,r)$, $1<r<p<\infty$; this yields a positiveanswer in the $1<p<2$ case to a conjecture of Stein which asserts that for$1<p<\infty$ we have that $C_d$ is of strong type $(p,p)$. A key ingredient inthis proof is the further extension of the {\it relational} time-frequencyperspective (introduced in \cite{q}) to the general polynomial phase.

On Stein's Conjecture on the Polynomial Carleson Operator