Recall that $(a;\,q)_\infty$ is the [$q$-Pochhammer symbol](http://mathworld.wolfram.com/q-PochhammerSymbol.html):
$$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$
Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes called the [Euler function](http://mathworld.wolfram.com/EulerFunction.html). It appears in Euler's [pentagonal number theorem](http://mathworld.wolfram.com/PentagonalNumberTheorem.html), and its reciprocal $(q;\,q)_\infty^{-1}$ is the generating function for the [partition numbers](http://mathworld.wolfram.com/PartitionFunctionP.html). It is also related to [Jacobi theta functions](http://mathworld.wolfram.com/JacobiThetaFunctions.html) and [Ramanujan theta functions](http://mathworld.wolfram.com/RamanujanThetaFunctions.html).

Let
$$f(x) = \frac{(q;\,q)_\infty}{(-q;\,-q)_\infty},\quad\text{where}\,\,q=e^{-\pi\sqrt x}.\tag2$$
In the OEIS entry [A080054](http://oeis.org/A080054) there is an empirical observation by Simon Plouffe that apparently $f(1)=\sqrt[8]2$. 

>**Empirical** (Simon Plouffe, Feb. 20, 2011):
$$\sum_{n=1}^{\infty}e^{-\pi(n-1)}a(n) =\sqrt[8]2.$$

I did some numerical experiments related to this observation, and the outcomes suggest a fascinating stronger conjecture:

_Conjecture:_ For every $p\in\mathbb Q,\,p>0$, the value $f(p)$
is an algebraic number.

For example, it appears that
$$f(3/5) = \sqrt[8]{2} \cdot \sqrt[4]{9 \sqrt{5}+5 \sqrt{15}-11 \sqrt{3}-19},$$
and $f(13/7)$ is an algebraic number of degree $96$ whose minimal polynomial is
$$x^{96}-647442063456 \, x^{88}+16702438371168 \, x^{80}-529345497357824
   \, x^{72}+4159684203040512 \, x^{64}-12099397290541056
   \, x^{56}+16408771708010496 \, x^{48}-10607690933600256
   \, x^{40}+2651923007078400 \, x^{32}-367001600 \, x^{24}+257949696
   \, x^{16}-100663296 \, x^8+16777216$$
and an isolating rational interval is $(37/36,\,6/5)$.

Is this conjecture new? Is it known to be true? If not, can you suggest any ideas how to (dis-)prove it?