In Mean values of multiplicative functions over function fields the mention a proof of Halasz inequality in one of their future pre-prints. In fact here is an a proof from 1999.

I would like to see a proof in any way shape or form, as it might lead to a proof of the prime number theorem.

There are various statements floating around here are two:

[1,3] Let $f : \mathbb{N} \to \mathbb{C}$ be multiplicative function such that $|f(n)| \leq 1$. Then if we establish for all real numbers $|t| < T$ that: $$ \sum_{p < x} \frac{1 - \mathrm{Re}[\, f(p) p^{-it} \,]}{p} \geq M$$ then the average value of $f(n)$ is quite small: $$ \frac{1}{x} \sum_{n < x} f(n) \ll (1+M)e^{-M} + \frac{1}{\sqrt{T}} $$ even the "cheap" version stated in the blog might already imply the prime number theorem.

[2] If the mean value of $f$ is "large" in absolute value, then $f(n)$ pretends to be $n^{it}$ for some "small" real $t$. Here, pretending is defined in terms of Kullback-Liebler distance:

$$ \mathbb{D}(f, g; x)^2 = \sum_{p \leq x} \frac{1 - \mathrm{Re}\big[\,f(p)\overline{g(p)}\,\big]}{p} $$

All of the proofs seem long and technical, and the new proof may not be much simpler. I still might like to know if the weaker version implies PNT