How do I suppress an automatic sign change? II

Let me be try to be highly specific, as my previous attempt How do I suppress an automatic sign change? to pose the question initially had a sign error, and perhaps became a little muddled.

---

---

In the course of pursuing the question Evaluate a certain three-dimensional constrained integral, the term (one of 694)

r = (202338335476512488921084723200 x^6 Sqrt[-(-1 + 2 x) (2 - x +  2 Sqrt[1 - x - 2 x^2])]Boole[1/38 (10 - Sqrt[5]) < x <= 1/4])/(319794090309 (723 + 17 Sqrt[5]))

is generated.

My attempt, r/.c, to apply (so the term becomes integrable–as can be checked) the rule (now corrected from earlier version–again, my apologies)

c := Sqrt[-(-1 + 2 x) (2 - x + 2 Sqrt[1 - x - 2 x^2])] -> 1 - 2 x + Sqrt[1 - x - 2 x^2]

fails (because apparently the expression -(-1 + 2 x) is ab initio converted to (1-2 x)).

What needs to be done, so that the intended conversion takes place?

Unfortunately, it would seem the apparent "automatic" conversion of $-(-1 + 2 x)$ to $(1-2 x)$ is not so "automatic" that it is performed in the formula for $r$ itself, which would obviate the apparent dilemma.

denestSqrt[e_, domain_, x_] := Replace[
    y /. Solve[Simplify[Reduce[Reduce[y == e && domain, x], y, Reals], domain], y],
    {
    {r_} :> r,
    _ -> e
    }
]

r /. c

r /. Sqrt[s_] :> denestSqrt[Sqrt[s], 1/38 (10 - Sqrt[5]) < x <= 1/4, x]