Is the function for the Weak Goldbach Conjecture an increasing function?

Question

Premise

There is a function used to count the number of ways a given odd integer larger than 5 can be written as the sum of three prime numbers. I have seen the function ($f_{3}$) expressed as the following in papers (or discussions) for the Weak Goldbach Conjecture.

$$ f_{3}(x) = \int_{R/Z} S(x,\, \alpha) e(-x\alpha) \, d\alpha $$ $$ e(\theta) := e^{2\pi i \theta} $$ $$ S(x,\, \alpha) := \sum_{p \leq x} e(\alpha p) $$

Question

I had been looking at this function and wanting to understand its characteristics. Is this function an increasing function?

Meaning, is the following true?

$$ f_{3}(x+2) - f_{3}(x) \geq 0 $$

Thoughts

I feel like it is an increasing function because the function $S$ either has the same number of primes/terms or more. But, it looks like $e(-x\alpha)$ could imply that the function could be decreasing or at least not be an increasing function.

Answer 1

https://oeis.org/A054860 gives "Number of ways of writing $2n+1$ as $p + q + r$ where $p, q, r$ are primes with $p \le q \le r$." It starts $0, 0, 0, 1, 2, 2, 2, 3, 4, 3, 5, 5, 5, 7, 7, 6, 9, 8, 9, 10, 11, 10, 12, 13, 12, 15, 16$, so you can see it's not an increasing function. E.g., $f(17)=4$ but $f(19)=3$.

Answer 2

For odd $x$, it is known due to I. M. Vinogradov that for odd $x\ge7$ we have

$$ f_3(x)={x^2\over2\log^3x}\prod_{p|x}\left(1-{1\over(p-1)^2}\right)\prod_{p\nmid x}\left(1+{1\over(p-1)^3}\right)+O\left(x^2\over\log^4x\right). $$

From the infinite product factors, it is certain that $f_3(x)$ cannot be monotonic for large $x$ either.