Suppose we have a Wiener stochastic integral $$\int_{t-h}^t f(r_v) d W_v, \tag{1}$$ It is well known that, by the Ito isometry property, $$\mathbb E \left [\left(\int_{t-h}^t f(r_v) d W_v\right)^2 \bigg | \mathcal F_{t-h}\right] = \int_{t-h}^t \mathbb E[f(r_v)^2|\mathcal F_{t-h}] d v.$$
I am wondering how to extend this result to fourth conditional moments in general $$\mathbb E \left[\left(\int_{t-h}^t f(r_v) d W_v\right)^4 \bigg | \mathcal F_{t-h}\right].$$
There is an answer to a similar question here that implies: $$ \mathbb E \left[\left(\int_{t-h}^t f(r_v) d W_v\right)^4 \bigg| \mathcal F_{t-h}\right] = 3 \left[\int_{t-h}^t \mathbb E[f(r_v)^2 |\mathcal F_{t-h}] d v\right]^2 \tag{2} $$
However, it seems that the answer relies on the fact that $\mathbb E (Y^4) = 3 (\sigma^2)^2$ for any $Y \sim N(0,\sigma^2).$
So my question goes, how can I compute the conditional fourth moment of (1) when $\int_{t-h}^t f(r_v) d W_v$ is not normally distributed? Should I expect result (2)?
An example with $f(x) = x$ would be computing $$\mathbb E \left[ \left(\int_{t-h}^t r_v d W_v\right)^4 \bigg | \mathcal F_{t-h} \right]$$ given, e.g. $$d r_{t} = \mu(r_t) d t + \eta \sqrt{r_t} d W_t.$$
Thank you.
This falls into studying multiple iterated Wiener integrals (see Wiener chaos). Here one can at least get an upper bound. As mentioned in "Notes on the Itô Calculus" by Steven P. Lalley
and so for $m=2$ we get
$$E\left(\int f dW\right)^{4}\leq c_{2}E\left[\left(\int f^{2} dW\right)^{2}\right]^{2}=c_{2}\left(\int Ef^{2} ds\right)^{2}.$$
In case the link dies, the proof simply uses that Hermite polynomials $H_{2m}(x,t)$ of Itô integrals are martingales and that the leading coefficient is $x^{2m}$.