Improved lower bounds for moments of Riemann zeta function

I am working on moments of the Riemann zeta function, and want to get a numerical lower bound for the $k$th moment of $\zeta(s)$ of the form $$\int_1^T|\zeta(1/2+it)|^{2k}dt>C(k)T(log T)^{k^2}.$$

Soundararajan (https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300011438) obtained a constant $C(k)=2$ for all $k$. In the recent preprint https://arxiv.org/pdf/2606.27323 an unconditional lower bound for the sixth moment of the Riemann zeta function was given: $$\int_1^T|\zeta(1/2+it)|^6dt>34c_3T(log T)^9,$$ with the conjecture being an asymptotic equal to $42c_3T(log T)^9$.

My question is:

Can the work of https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300011438 give a constant better than $2$? Even an answer depending on $k$ would be good.

Can the method of https://arxiv.org/abs/2606.27323 be adapted to give a stronger bound?

I am mostly interested in the numerical constants, we know the order of magnitude, but I want to find the best possible numerical bounds, because I want to see how close to the conjectures of Keating-Snaith one can unconditionally get.