TITLE: A conjecture about algebraic values of $(-q;\,-q)_\infty/(q;\,q)_\infty$
QUESTION [14 upvotes]: Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol:
$$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$
Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes called the Euler function. It appears in Euler's pentagonal number theorem, and its reciprocal $(q;\,q)_\infty^{-1}$ is the generating function for the partition numbers. It is also related to Jacobi theta functions and Ramanujan theta functions.
Let
$$f(x) = \frac{(-q;\,-q)_\infty}{(q;\,q)_\infty},\quad\text{where}\,\,q=e^{-\pi\sqrt x}.\tag2$$
In the OEIS entry A080054 there is an empirical observation by Simon Plouffe that apparently $f(1)=\sqrt[8]2$. 

Empirical (Simon Plouffe, Feb. 20, 2011):
  $$\sum_{n=0}^{\infty}e^{-\pi n}a(n) =\sqrt[8]2.$$

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?

REPLY [2 votes]: This is not meant an answer, instead I wish to list some computational values for $f(x)$. Not sure if these are known.
$$f(1)=\sqrt[8]2\,,$$
$$f(2)=\sqrt[16]2\,\sqrt[4]{\cos\frac{\pi}8}\,, \qquad
f(4)=\sqrt[16]2\,\sqrt{\cos\frac{\pi}8}\,,$$
$$f(3)=\sqrt[4]{\sec\frac{\pi}{12}}\,, \qquad 
f(1/3)=\sqrt[4]{\csc\frac{\pi}{12}}\,,$$
$$f(1/2)=\sqrt[4]{1+\sqrt2}\,, \qquad
f(1/4)=\sqrt{1+\sqrt2}\,\,.$$