The following conjecture which I posed has been verified for all $n<10000$ , my question is to find a proof or a disproof of it. The conjecture is as follows: Consider for all $n \in \mathbb N$ , $$ a(n)=\binom{\operatorname{prime}[n+1]}{\operatorname{prime}[n]} $$ and $$ b(n)=\frac{\operatorname{prime}[n]^{\operatorname{prime}[n+1]}}{\operatorname{prime}[n+1]^{\operatorname{prime}[n]}}, $$ where $\operatorname{prime}[m]$ is the $m$-th prime number); then for all $n \in \Bbb N$, $$ a(n+1)>a(n)\:\text{ if and only if }\:b(n+1)>b(n). $$
Example: take $n=4$ ,$\operatorname{prime}[4]=7$ and $\operatorname{prime}[5]=11$ and $\operatorname{prime}[6]=13$. Then $$ \binom{13}{11} < \binom{11}{7}\:\text{ and also }\:\frac{11^{13}}{13^{11}} < \frac{7^{11}}{{11^7}} $$ A further example that shows that the conjecture is not trivial: in some cases among composite numbers the comparison test fails, for example take the numbers $35$ , $60$ , $78$ , we can notice that $$ \binom{78}{60}>\binom{60}{35}\:\text{ but }\:\frac{60^{78}}{78^{60}}<\frac{35^{60}}{60^{35}}. $$ Thank you.