An ergodic theory problem

Let $$T (x)= 4x (1 − x)$$ be a map from $$X = [0, 1]$$ into itself. Prove or disprove that this map has a trajectory of period $$7$$.

Note that $$T(\sin^2 t) = 4\sin^2t\cos^2 t=\sin^2(2t)\ ,$$ so that \begin{aligned} T^{(7)}(\sin^2 t) &:= ( T\circ T\circ T\circ T\circ T\circ T\circ T) (\sin^2 t) \\ &=\sin (2^7 t)\ , \end{aligned} and we can now easily find a fixed point, by specifying the difference between $$t$$ and $$2^7t$$ to be some small multiple of $$2\pi$$, for instance by requiring $$2^7 t-t=2\pi$$. This gives $$t=2\pi/127$$. (Or the difference may be $$4\pi$$, or $$6\pi$$, or ...) Fixed points of period seven are thus $$\sin ^2\frac {2k\pi}{127}\ ,\qquad k=1,2,\dots, 126\ .$$

Numerically, pari/gp code:

? T(x) = 4 * (1-x) * x;
? p(x) = T(T(T(T(T(T(T(x))))))) - x;
? p( sin(2*Pi/127)^2 )
%8 = 1.6071211827648462023 E-40
? p( sin(4*Pi/127)^2 )
%9 = 6.428484731059384809 E-40
? p( sin(6*Pi/127)^2 )
%10 = 2.755064884739736347 E-38


Here is also an exact check working in the cyclotomic field of order $$4\cdot 127$$, which contains $$j=\sqrt {-1}$$ and a primitive root $$z$$ of unity of order $$127$$:

sage: K.<z> = CyclotomicField(127*4)
sage: R.<x> = PolynomialRing(K)
sage: w = z^4    # so w is a primitive root of unity of order 127
sage: j = z^127  # so j is a primitive root of unity of order 4, it is sqrt(-1)
sage: w.multiplicative_order(), j.multiplicative_order()
(127, 4)
sage: T = 4*x*(1-x)
sage: p = T(T(T(T(T(T(T(x))))))) - x
sage: p( ( (w - 1/w) / (2*j) )^2 )    # corresponds to p( sin(2pi/127)^2 )
0
sage: p( ( (w^2 - 1/w^2) / (2*j) )^2 )    # corresponds to p( sin(4pi/127)^2 )
0
sage: set( [ p( ( (w^k - 1/w^k) / (2*j) )^2 ) for k in [1..126] ] )
{0}


The last set containing the only element zero tells us, that all values listed above, $$\sin^2(2k\pi/127)$$ are roots of $$p(x)=T^{(7)}(x)-x$$. (Of course, we avoid $$0,3/4$$, which are the (only) fixed points of $$T$$