(This answer uses ancillae and feedback)
Does anyone know if there has been any improvement on the decomposition of the general n-qubits control X with phase differences in terms of elementary gates up to this day? [...] And also what is the theoretical lower bound?
About a week ago I would have told you it's probably not possible to do better than a T cost of $4n \pm O(1)$. But then, well...
Here's a construction that reduces an $n$-control Toffoli into an $n-2$ control Toffoli, with a 50/50 chance of introducing heralded phase error on the controls. It uses 6 T gates, 1 ancilla, 12 stabilizer gates, and classical feedback.
By iterating this construction, you can perform an $n$-control Toffoli with relative phase error using $3n$ T gates, $n/2$ ancillae, and $6n$ stabilizer gates.
The downside is that you will eventually need to correct the relative phase error, and this will cost you more than $n$ T gates so you ultimately come out behind vs just using ANDs to temporarily merge controls which has a total cost (including eventually fixing the phase error) of $4n$ T gates (that construction is explained in halving the cost of quantum addition).