The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

With respect to the recent breakthrough, Bourgain states in this preprint:

Concerning applications to the zeta-function, our work as it stands does not lead to further progress. The reason for this is that we did not explore the effect of large $k$ (possibly depending on $N$) and in the present form is likely very poor. A similar comment applies to Wooley’s approach

This is also discussed in Ford's papers Vinogradov's Integral and Bounds for the Riemann Zeta Function, Zero-Free Regions for the Riemann Zeta Function, where he states:

Lastly we indicate what the limit of our method is, i.e. what could be accomplished with [a version of Vinogradov's Mean Value Theorem]... [this] yields Theorem $1$ with a constant $B=\sqrt{2}+\epsilon$ (valid for $\sigma\geq\frac{15}{16}$) where $\epsilon$ can be taken arbitrarily small.

Here Theorem $1$ is the inequality $$\zeta(\sigma+it)\leq A t^{B(1-\sigma)^{3/2}}\log^{2/3}t\ \ \ \ \ \ (t\geq 3,\ \frac{1}{2}\leq \sigma\leq 1)$$ which Ford proves with $A=76.2$ and $B=4.45$.

[For additional discussion on the implications of the Bourgain-Demeter-Guth breakthrough see, this recent preprint of Heath-Brown A New $k^{th}$ Derivative Estimate for Exponential Sums via Vinogradov’s Mean Value]

And a few words about "possible generalizations". The main object here is the sum $$S=\sum_{u\asymp a}e^{2\pi i F(u)},$$ where $F(u)=-\frac{t\log u}{2\pi}$, $t=a^{n-\theta}$, $0\le \theta<1$. It is known that $|S|\ll a^{1-\frac{c}{n^2}}$. It was mentioned by Vinogradov in his book "Trigonometrical sums in number theory" that even the much stronger estimate $|S|\ll a^{1-\frac{c}{n}}$ (unreachable by this method) will give only $$\pi(x)=\text{li}(x)+O\left(x\exp\left(c\log(x)^{2/3}(\log \log x)^{-1/5}\right)\right).$$