Upper bounds on $E[{\rm var}^2(X|Y)]$

Question

I am looking for an upper bound on the quantity \begin{align*} E[{ \rm var}^2(X|Y)] \end{align*}

where ${\rm var}(X|Y)=E[(X-E[X|Y])^2|Y]$.

Getting a lower bound is rather easy using Jensen's inequality \begin{align*} E[{ \rm var}^2(X|Y)] \ge E^2[{ \rm var}(X|Y)]= {\rm MMSE}^2(X|Y) \end{align*}

If general upper bound does not exist then we can assume that $X$ is zero mean unit varience and $Y=X+Z$ where $Z\sim \mathcal{N}(0,1)$ and independent of $X$. As an example, if $X \sim \mathcal{N}(0,1)$ then we can actually compute $E[{ \rm var}^2(X|Y)]$ which is given by \begin{align*} { \rm var}^2(X|Y)=\frac{1}{4} \end{align*}

Answer 1

\begin{align}E[\rm{var}^2(X\mid Y)] := & E\left[\left(E[(X-E[X|Y])^2\mid Y]\right)^2\right]\\ \leq & E\left[E[(X-E[X|Y])^4\mid Y]\right] \\= & E[(X-E[X|Y])^4]\end{align}

The first inequality comes from Jensen applied on the conditional expectation inside. The last equality comes from iterated conditioning.