Can there be functions $g$ and $f$ such that $\rho_{f(X),g(X+Y)}^2 > \rho_{X,X+Y}^2$ Motivation a special case
A special case of this question is an inequality between the Spearman's rank correlation and the Pearson correlation (Why is the sum of individual Spearman's rho squared less than 1 as opposed to Pearson's r in a synthetic example? ). The expectation of the squared sample Spearman's rank correlation will approach $\rho_{f(X),g(X+Y)}^2$ with $f$ and $g$ the cumulative distribution function of the variables $X$ and $X+Y$.
By trying out several different distributions for $X$ and $Y$ it seems that in all cases the Spearman's rank correlation coefficient is smaller than the Pearson's correlation coefficient.
Thinking about that question it seemed to me intuitive that the correlation between $X$ and $X+Y$ must be a maximum. When we apply a formula, then information gets lost and the correlation should decrease. I imagine that a correlation between $f(X)$ and $g(X+Y)$ can't be larger than a correlation between the original $X$ and $X+Y$.
General question
So I am wondering whether, beyond that example with the Spearman's rank correlation it is true in general for other functions as well:

If $X$ and $Y$ are independent, then the correlation should decrease or at least stay the same when we apply functions to $X$ and $X+Y$.  $\rho_{f(X),g(X+Y)}^2 \leq \rho_{X,X+Y}^2$

Is this true?
Is there a simple proof for this?

Edit: Apparently there can be made many examples relatively easy. Could it also work when $Var(Y) > Var(X)$ and $f$ and $g$ are monotonic functions?

An example:
A distribution $X$ with values $-1$ and $1$ and a distribution $Y$ with values $-0.5$ and $0.5$ (e.g. scaled Rademacher distributed variables).
Then let $g$ and $f$ be a sign function
$$g(x) = f(x) = \begin{cases}
 -1 & \quad \text{if $x<0$}\\
 0 & \quad \text{if $x=0$}\\
 1 & \quad \text{if $x>0$}\end{cases}$$
This makes $f(X) = g(X+Y)$ and makes the correlation $1$.


Edit: Apparently there can be made many examples relatively easy. Could it also work when $Var(Y) > Var(X)$ and $f$ and $g$ are monotonic functions?

We can adapt the example from statmerkur and use $X \sim Exp(1)$ and $Y \sim N(0,4)$ and the functions $f(x) = g(x) = x^3$.
