Limit of conditional expectations (when limit linked to the conditionning)

I am working with conditional expectations and am trying to derive a limit property.

Consider $(Y_n)_{n in mathbb{N}}$ a sequence of square integrable random variables, that converge in $L^2$ to a square integrable random variable $Y$. Additionally assume that $mathbb{E}[Y_n|Y] = Y_n$ (for example, $Y_n$ is a sequence of discrete quantizers of $Y$).

Is there anyway at all of guaranteeing that for some other $X$ in $L^2$, and for some form of convergence ($L^2$, $mathbb{P}$ etc.) :

$$ lim_{n rightarrow + infty} mathbb{E}[X|Y_n] = mathbb{E}[X|Y]. $$

I am aware of the following similar question :

https://math.stackexchange.com/questions/3096326/conditional-expectation-of-asymptotically-independent-random-variables

but in that case $mathbb{E}[Y_n|Y] = Y_n$ does not hold…

With a $L^2$-projection approach to conditional expectation, and with $Y_n$ converging in $L^2$ to $Y$, I keep thinking there must be some way of getting this to work… But maybe it just won’t.

Thank you for any suggestions !