It is stated in several places, as wikipedia, or oeis, or MathWorld that the ratio of two consecutive superior highly composite numbers is a prime, and all of them give as the only reference a paper by Ramanujan:
Ramanujan, S. "Highly Composite Numbers." Proc. London Math. Soc. 14, 347-407, 1915.
On the other hand, the analogous result for colosally abundant numbers is presented as a conjecture. In fact, Alaoglu and Erdős prove that this is true assuming that any $\alpha$ satisfying that $p^\alpha, q^\alpha$ are rational numbers for two prime numbers $p\neq q$ must be an integer. This is a consequence of the four exponentials conjecture.
However, when looking at the Ramanujan's proof, I cannot see any difference between both cases. Namely, Ramanujan says (page 392) that "it is easily seen" from (195) and (196). The first reference states that a superior highly composite number $N_x$ is a product of primorials $p_1\#\cdots p_k\#$, where $p_r$ is the greatest prime not exceeding $(1+1/r)^x$, where $x=1/\epsilon$ and $\epsilon$ is the epsilon appearing in the definition of superior highly composite number.
The second reference is an equivalent formulation, namely, that $N=e^{\vartheta(2)^x+\vartheta(3/2)^x+\cdots }$
It seems to me that the situation is exactly the analogous one to that considered by Alaoglu and Erdős: the change from a number $N_x$ to the next one happens when some $(1+1/r)^x$ takes a prime value, and if this happens for two different $r$ at the same $x$, i.e., if $(1+1/r)^x=p$, $(1+1/s)^x=q$, then $p^{1/x}, q^{1/x}$ are rational numbers, and this leads to a contradiction if we can conclude that $1/x$ is an integer.
So, my question is: is there really an argument proving that the ratio of two consecutive superior highly composite numbers is a prime not relying in the above-mentioned unproven conjecture?
Is seems that a negative answer is implicit in the answer to this question, but it seems strange to me that the statement is still be presented everywhere as a proven fact.
