Number of real roots of an iterated quadratic: $x^2-3/2$

I was messing around with polynomials and their real roots when I, as recreational mathematicians do, asked myself the following random question:

Suppose I am given a polynomial $P(x)$. How can I find the number of real roots of the polynomial $P^{\circ n}(x)$, representing the n-fold composition of $P$ with itself (counting multiplicity)?

I started with the simple polynomial $P_1(x)=x^2-1$. This was an easy example, as it turned out that $P_1^{\circ n}(x)$ has $n+1$ real roots, which was simple to prove.

My next example was the polynomial $P_2(x)=x^2-2$. This one was more difficult, but I eventually determined that $P_2^{\circ n}(x)$ has $2F_{n+1}-2$ real roots, where $F_n$ represents the sequence of Fibonacci numbers with $F_0=F_1=1$.

In general, I am considering polynomials of the form $P_c(x)=x^2-c$. For $c$ less than $-1$, the iterates of this polynomial have no real roots, and for $c$ greater than $3$, the $n$th iterate of this polynomial seems to have $2^n$ real roots.

The polynomial with which I am stumped is $$P_{3/2}(x)=x^2-\frac{3}{2}$$ While I have been unable to derive a formula for the number of real roots of $P_{3/2}^{\circ n}(x)$, by observing the number of real roots for the first couple of iterations, I have come up with the remarkable conjecture that the number of real roots of the $n$th iterate is given by $2p(n)$, where $p(n)$ represents the number of partitions of $n$.

This conjecture, if true, would be truly amazing. How can I prove it?

NOTE: To prove the formulae I obtained for $P_1$ and $P_2$, I divided the real line up into intervals that the polynomials in question sent to one another, and from this I obtained a recursive formula for each. However, I cannot figure out how to nearly divide $\mathbb R$ into intervals in the same way.

I am unable to reproduce your results for $c=2$. I evaluate $P_2^{\circ 4}(x)$ as $x^{16} - 16x^{14} + 104x^{12} - 352x^{10} + 660x^8 - 672x^6 + 336x^4 - 64x^2 + 2$, which has $16$ real roots rather than $2F_5-2 = 14$.

As for $c=\frac32$, I confirm your coincidence up to $n=8$, but after that it breaks down. I calculate the first ten values as 2, 4, 6, 10, 14, 22, 30, 44, 58, 82.