Failure of weak mixed-norm inclusion into a sum space

Let $k$ and $m$ be parameters satisfying $1\leq k<+\infty$ and $1<m<+\infty$, and let $p,q,p',q'$ satisfy

$$\frac{k}{p}+\frac{m}{q}=1,$$ $$\frac{k}{p'}+\frac{m}{q'}=1,$$

with $m\leq q'<q<+\infty$. The weak mixed-norm non-inclusion conjecture. One has

$$L_T^{p,\infty}L_x^{q,\infty}\not\subset L_T^{p'}L_x^{q'}+L_T^kL_x^\infty.$$

This concerns whether weak control in both time and space can be decomposed into the corresponding strong mixed-norm space and an $L_T^kL_x^\infty$ component; the asserted non-inclusion is presented as an open question in the paper.