Lexi-bounds conjecture for short linear codes and saturating sets

Let $\ell_q(r,3)$ be the length function, let $\ell_q(r,3,5)$ be the $d$-length function, let $s_q(r-1,2)$ be the smallest size of a $2$-saturating set in $\mathrm{PG}(r-1,q)$, and let $s_q^{\operatorname{arc}}(3)$ be the smallest size of a complete arc in $\mathrm{PG}(3,q)$. Lexi-bounds conjecture. The following upper bounds hold:

$$\ell_q(4,3)=s_q(3,2)\leq\ell_q(4,3,5)=s_q^{\operatorname{arc}}(3)<2.8\sqrt[3]{\ln q}\,\sqrt[3]{q}$$

for all $q\geq 11$, and

$$\ell_q(5,3)=s_q(4,2)\leq\ell_q(5,3,5)<3\sqrt[3]{\ln q}\,\sqrt[3]{q^2}$$

for all $q\geq 5$. These bounds improve earlier constants and connect short linear codes with saturating sets and complete arcs; the source presents them as conjectural bounds based on computational observations.