Concerning 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 briefly 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:

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]

The short answer is given in this recent preprint of Bourgain:

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 briefly 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 more 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:

The purpose of this paper is to show how a slightly refined version of the original method of Vinogradov leads to distinctly stronger bounds. However it needs to be emphasised from the start that the power of these new results derives very largely from new estimates for the Vinogradov mean value integral, due to Wooley (for $l = 3$), and to Bourgain, Demeter and Guth (for $l\geq 4$).

Concerning 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 briefly 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: