Is the term $E\left[Z^3E[Z\mid Y] \right]$ positive or negative?

Question

Let $Z$ be standard normal and $Y$ some other random variable.

Is the $E\left[Z^3E[Z\mid Y] \right]$ positive or negative? I tried a few things like using orthogonality principle \begin{align} E\left[Z^3E[Z\mid Y] \right]=E[E[Z^3\mid Y]E[Z\mid Y]]=E[E[Z^3\mid Y]Z] \end{align}

but this didn't go anywhere?

This question is related to something I asked a while ago here. But now the structure of $Y$ is unspecified.