I question the use of the term "conjecture," which, as explained

here, implies the statement is important and worthy of study. The term "claim" appears to be more appropriate. Offhand, I see no reason to consider it, even if correct, as anything other than a curiosity.

The claim is also poorly stated. The phrase "if x is sufficiently large" is entirely superfluous. The definition of asymptotic equality involves the limit as x increases without bound, or tends to plus infinity.

Your request indicates to me that your series is difficult to evaluate. There are already expressions proven to be asymptotically equal to the prime-counting function, such as li(x) and x/log(x), which are easy to evaluate.