Upper and Lower Bounds on $Var(Var(X\mid Y))$

Are there any particular properties that \begin{align*} Var(Var(X\mid Y)) \end{align*} satisfies so that we can derive any upper and lower bounds on it. For example, if we replace $Var$ with expectation we have \begin{align*} E[E[X\mid Y]]=E[X] \end{align*}

This question is somewhat related to the one found here.

One way to proceed is to use \begin{align*} Var(Var(X\mid Y))&=E[Var^2(X\mid Y)]-(E[Var(X\mid Y)])^2\\ &=E[Var^2(X\mid Y)]-MMSE^2(X|Y)\\ \end{align*}

where we can bound $E[Var^2(X\mid Y)]$ as \begin{align*} E[Var^2(X\mid Y)]&=E[(E[(X-E[X|Y])^2|Y])^2] \\ &\le E[(E[(X-E[X|Y])^4|Y)]\\ &= E[(X-E[X|Y])^4] \end{align*} and we have a bound

\begin{align*} Var(Var(X\mid Y)) &\le E[(X-E[X|Y])^4]-MMSE^2(X|Y)\\ &=E[(X-E[X|Y])^4]-(E[(X-E[X|Y])^2])^2 \end{align*}

The question is can we do better and find a tighter bound?

If you need further assumption we can assume that $Y=X+Z$ where $X$ and $Z$ are finite variance and zero mean and independent.

I would be very grateful for any ideas you guys might have?