On a product involving Ramanujan primes We denote the $k$th Ramanujan prime as $\mathcal{R}_k$, that is the sequence A104272 from the OEIS as you can read from this Wikipedia. Then I was inspired in Richard K. Guy, Unsolved Problems in Number Theory, Springer, Volume I (1994), section B48 p. 102, to write the following conjecture*. In the Question below I am asking what can be elucidate about it. 
By definition we know that there exist infinitely many Ramanujan primes.
Conjecture. For $n>1$ $$\prod_{k=1}^n\frac{\mathcal{R}_k+1}{\mathcal{R}_k-1}\tag{1}$$
is never an integer.

Question. My belief is that there exists an integer $n_0$ such that $(1)$ is never an integer $\forall n>n_0$. What work can be done about previous Conjecture or my belief? Many thanks.


*We can state similar conjectures that harmonize with previous one involving different prime constellations: if we write the $k$th term of the set of lesser of twin primes, I am saying A001359 from the OEIS, denoted as $t_k$, instead of $\mathcal{R}_k$, or well if we write the $k$th Sophie Germain prime, A005384, denoted as $\mathcal{G}_k$, instead of $\mathcal{R}_k$.
 A: I can now say with some certainty that this conjecture holds for the first 500 Ramanujan primes (though my computer has checked to some number above that $\leq 600$), which is pretty large (8831), though I don't have the fastest programs in the world.
Also, see my comment in reply to @didgogns, where I give a proof/proof-sketch to show that this product does indeed diverge. 
I shall now consider some heuristics...
EDIT I: However unrelated to the question, it occurs to me that what I proved in the comment section in fact shows that $$\prod^{n}_{k=1}\frac{P_n+m}{P_n-m}$$ diverges for any $m\in\mathbb{R}^+$, as long as the sum of the reciprocals of the sequence $P_n$ diverge (i.e. $P_n$ could be the primes, the Ramanujan primes etc.).
EDIT II: My program has now verified the conjecture for the first 700 Ramanujan primes, though with potentially less certainty than before, since floating-point errors become a bit more important at this level...
EDIT II: I realise that I made a mistake in my proof in the comment section, saying that the numerator of the sum of the reciprocals of $R_i$ would looke like $R_1+R_2+R_3...$, whereas it would instead look like the sum of all $k$-products (products of $k$ of the primes). This doesn't change the conclusion of the proof, however, as it is still true that $(R_1+1)(R_2+1)(R_3+1)...$ contains in its expansion all of these terms and many more (in fact, it contains the sum of all $i$-products, for all $1\leq i \leq k$).
EDIT III: This is probably highly unnecessary at this point, but my program has now verified this conjecture (with much less certainty) to the first 800 Ramanujan primes.
